Primitive. The name Eps_i is a term of type (setprop)set.
Axiom. (Eps_i_ax) We take the following as an axiom:
∀P : setprop, ∀x : set, P xP (Eps_i P)
Definition. We define True to be ∀p : prop, pp of type prop.
Definition. We define False to be ∀p : prop, p of type prop.
Definition. We define not to be λA : propAFalse of type propprop.
Notation. We use ¬ as a prefix operator with priority 700 corresponding to applying term not.
Definition. We define and to be λA B : prop∀p : prop, (ABp)p of type proppropprop.
Notation. We use as an infix operator with priority 780 and which associates to the left corresponding to applying term and.
Definition. We define or to be λA B : prop∀p : prop, (Ap)(Bp)p of type proppropprop.
Notation. We use as an infix operator with priority 785 and which associates to the left corresponding to applying term or.
Definition. We define iff to be λA B : propand (AB) (BA) of type proppropprop.
Notation. We use as an infix operator with priority 805 and no associativity corresponding to applying term iff.
Beginning of Section Eq
Variable A : SType
Definition. We define eq to be λx y : A∀Q : AAprop, Q x yQ y x of type AAprop.
Definition. We define neq to be λx y : A¬ eq x y of type AAprop.
End of Section Eq
Notation. We use = as an infix operator with priority 502 and no associativity corresponding to applying term eq.
Notation. We use as an infix operator with priority 502 and no associativity corresponding to applying term neq.
Beginning of Section FE
Variable A B : SType
Axiom. (func_ext) We take the following as an axiom:
∀f g : AB, (∀x : A, f x = g x)f = g
End of Section FE
Beginning of Section Ex
Variable A : SType
Definition. We define ex to be λQ : Aprop∀P : prop, (∀x : A, Q xP)P of type (Aprop)prop.
End of Section Ex
Notation. We use x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using ex.
Axiom. (prop_ext) We take the following as an axiom:
∀p q : prop, iff p qp = q
Primitive. The name In is a term of type setsetprop.
Notation. We use as an infix operator with priority 500 and no associativity corresponding to applying term In. Furthermore, we may write xA, B to mean x : set, xAB.
Definition. We define Subq to be λA B ⇒ ∀xA, x B of type setsetprop.
Notation. We use as an infix operator with priority 500 and no associativity corresponding to applying term Subq. Furthermore, we may write xA, B to mean x : set, xAB.
Axiom. (set_ext) We take the following as an axiom:
∀X Y : set, X YY XX = Y
Axiom. (In_ind) We take the following as an axiom:
∀P : setprop, (∀X : set, (∀xX, P x)P X)∀X : set, P X
Notation. We use x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using ex and handling ∈ or ⊆ ascriptions using and.
Primitive. The name Empty is a term of type set.
Axiom. (EmptyAx) We take the following as an axiom:
¬ ∃x : set, x Empty
Primitive. The name is a term of type setset.
Axiom. (UnionEq) We take the following as an axiom:
∀X x, x X ∃Y, x Y Y X
Primitive. The name 𝒫 is a term of type setset.
Axiom. (PowerEq) We take the following as an axiom:
∀X Y : set, Y 𝒫 X Y X
Primitive. The name Repl is a term of type set(setset)set.
Notation. {B| xA} is notation for Repl Ax . B).
Axiom. (ReplEq) We take the following as an axiom:
∀A : set, ∀F : setset, ∀y : set, y {F x|xA} ∃xA, y = F x
Definition. We define TransSet to be λU : set∀xU, x U of type setprop.
Definition. We define Union_closed to be λU : set∀X : set, X U X U of type setprop.
Definition. We define Power_closed to be λU : set∀X : set, X U𝒫 X U of type setprop.
Definition. We define Repl_closed to be λU : set∀X : set, X U∀F : setset, (∀x : set, x XF x U){F x|xX} U of type setprop.
Definition. We define ZF_closed to be λU : setUnion_closed U Power_closed U Repl_closed U of type setprop.
Primitive. The name UnivOf is a term of type setset.
Axiom. (UnivOf_In) We take the following as an axiom:
∀N : set, N UnivOf N
Axiom. (UnivOf_TransSet) We take the following as an axiom:
∀N : set, TransSet (UnivOf N)
Axiom. (UnivOf_ZF_closed) We take the following as an axiom:
∀N : set, ZF_closed (UnivOf N)
Axiom. (UnivOf_Min) We take the following as an axiom:
∀N U : set, N UTransSet UZF_closed UUnivOf N U
Theorem. (andI)
∀A B : prop, ABA B
Proof:
An exact proof term for the current goal is (λA B a b P H ⇒ H a b).
Theorem. (orIL)
∀A B : prop, AA B
Proof:
An exact proof term for the current goal is (λA B a P H1 H2 ⇒ H1 a).
Theorem. (orIR)
∀A B : prop, BA B
Proof:
An exact proof term for the current goal is (λA B b P H1 H2 ⇒ H2 b).
Theorem. (iffI)
∀A B : prop, (AB)(BA)(A B)
Proof:
An exact proof term for the current goal is (λA B ⇒ andI (AB) (BA)).
Theorem. (pred_ext)
∀P Q : setprop, (∀x, P x Q x)P = Q
Proof:
Let P and Q be given.
Assume H1.
Apply func_ext set prop to the current goal.
Let x be given.
Apply prop_ext to the current goal.
We will prove P x Q x.
An exact proof term for the current goal is H1 x.
Definition. We define nIn to be λx X ⇒ ¬ In x X of type setsetprop.
Notation. We use as an infix operator with priority 502 and no associativity corresponding to applying term nIn.
Theorem. (EmptyE)
∀x : set, x Empty
Proof:
Let x be given.
Assume H.
Apply EmptyAx to the current goal.
We use x to witness the existential quantifier.
An exact proof term for the current goal is H.
Theorem. (PowerI)
∀X Y : set, Y XY 𝒫 X
Proof:
Let X and Y be given.
Apply PowerEq X Y to the current goal.
An exact proof term for the current goal is (λ_ H ⇒ H).
Theorem. (Subq_Empty)
∀X : set, Empty X
Proof:
An exact proof term for the current goal is (λ(X x : set)(H : x Empty) ⇒ EmptyE x H (x X)).
Theorem. (Empty_In_Power)
∀X : set, Empty 𝒫 X
Proof:
An exact proof term for the current goal is (λX : setPowerI X Empty (Subq_Empty X)).
Theorem. (xm)
∀P : prop, P ¬ P
Proof:
Let P of type prop be given.
Set p1 to be the term λx : setx = Empty P.
Set p2 to be the term λx : setx Empty P.
We prove the intermediate claim L1: p1 Empty.
We will prove (Empty = Empty P).
Apply orIL to the current goal.
An exact proof term for the current goal is (λq H ⇒ H).
We prove the intermediate claim L2: (Eps_i p1) = Empty P.
An exact proof term for the current goal is (Eps_i_ax p1 Empty L1).
We prove the intermediate claim L3: p2 (𝒫 Empty).
We will prove ¬ (𝒫 Empty = Empty) P.
Apply orIL to the current goal.
Assume H1: 𝒫 Empty = Empty.
Apply EmptyE Empty to the current goal.
We will prove Empty Empty.
rewrite the current goal using H1 (from right to left) at position 2.
Apply Empty_In_Power to the current goal.
We prove the intermediate claim L4: Eps_i p2 Empty P.
An exact proof term for the current goal is (Eps_i_ax p2 (𝒫 Empty) L3).
Apply L2 to the current goal.
Assume H1: Eps_i p1 = Empty.
Apply L4 to the current goal.
Assume H2: Eps_i p2 Empty.
We will prove P ¬ P.
Apply orIR to the current goal.
We will prove ¬ P.
Assume H3: P.
We prove the intermediate claim L5: p1 = p2.
Apply pred_ext to the current goal.
Let x be given.
Apply iffI to the current goal.
Assume H4.
We will prove (¬ (x = Empty) P).
Apply orIR to the current goal.
We will prove P.
An exact proof term for the current goal is H3.
Assume H4.
We will prove (x = Empty P).
Apply orIR to the current goal.
We will prove P.
An exact proof term for the current goal is H3.
Apply H2 to the current goal.
rewrite the current goal using L5 (from right to left).
An exact proof term for the current goal is H1.
Assume H2: P.
We will prove P ¬ P.
Apply orIL to the current goal.
We will prove P.
An exact proof term for the current goal is H2.
Assume H1: P.
We will prove P ¬ P.
Apply orIL to the current goal.
We will prove P.
An exact proof term for the current goal is H1.
Theorem. (FalseE)
False∀p : prop, p
Proof:
An exact proof term for the current goal is (λH ⇒ H).
Theorem. (andEL)
∀A B : prop, A BA
Proof:
An exact proof term for the current goal is (λA B H ⇒ H A (λa b ⇒ a)).
Theorem. (andER)
∀A B : prop, A BB
Proof:
An exact proof term for the current goal is (λA B H ⇒ H B (λa b ⇒ b)).
Beginning of Section PropN
Variable P1 P2 P3 : prop
Theorem. (and3I)
P1P2P3P1 P2 P3
Proof:
An exact proof term for the current goal is (λH1 H2 H3 ⇒ andI (P1 P2) P3 (andI P1 P2 H1 H2) H3).
Theorem. (and3E)
P1 P2 P3(∀p : prop, (P1P2P3p)p)
Proof:
An exact proof term for the current goal is (λu p H ⇒ u p (λu u3 ⇒ u p (λu1 u2 ⇒ H u1 u2 u3))).
Theorem. (or3I1)
P1P1 P2 P3
Proof:
An exact proof term for the current goal is (λu ⇒ orIL (P1 P2) P3 (orIL P1 P2 u)).
Theorem. (or3I2)
P2P1 P2 P3
Proof:
An exact proof term for the current goal is (λu ⇒ orIL (P1 P2) P3 (orIR P1 P2 u)).
Theorem. (or3I3)
P3P1 P2 P3
Proof:
An exact proof term for the current goal is (orIR (P1 P2) P3).
Theorem. (or3E)
P1 P2 P3(∀p : prop, (P1p)(P2p)(P3p)p)
Proof:
An exact proof term for the current goal is (λu p H1 H2 H3 ⇒ u p (λu ⇒ u p H1 H2) H3).
Variable P4 : prop
Theorem. (and4I)
P1P2P3P4P1 P2 P3 P4
Proof:
An exact proof term for the current goal is (λH1 H2 H3 H4 ⇒ andI (P1 P2 P3) P4 (and3I H1 H2 H3) H4).
Variable P5 : prop
Theorem. (and5I)
P1P2P3P4P5P1 P2 P3 P4 P5
Proof:
An exact proof term for the current goal is (λH1 H2 H3 H4 H5 ⇒ andI (P1 P2 P3 P4) P5 (and4I H1 H2 H3 H4) H5).
Variable P6 : prop
Theorem. (and6I)
P1P2P3P4P5P6P1 P2 P3 P4 P5 P6
Proof:
An exact proof term for the current goal is (λH1 H2 H3 H4 H5 H6 ⇒ andI (P1 P2 P3 P4 P5) P6 (and5I H1 H2 H3 H4 H5) H6).
Variable P7 : prop
Theorem. (and7I)
P1P2P3P4P5P6P7P1 P2 P3 P4 P5 P6 P7
Proof:
An exact proof term for the current goal is (λH1 H2 H3 H4 H5 H6 H7 ⇒ andI (P1 P2 P3 P4 P5 P6) P7 (and6I H1 H2 H3 H4 H5 H6) H7).
End of Section PropN
Theorem. (not_or_and_demorgan)
∀A B : prop, ¬ (A B)¬ A ¬ B
Proof:
Let A and B be given.
Assume u: ¬ (A B).
Apply andI to the current goal.
We will prove ¬ A.
Assume a: A.
An exact proof term for the current goal is (u (orIL A B a)).
We will prove ¬ B.
Assume b: B.
An exact proof term for the current goal is (u (orIR A B b)).
Theorem. (not_ex_all_demorgan_i)
∀P : setprop, (¬ ∃x, P x)∀x, ¬ P x
Proof:
Let P be given.
Assume H1.
Let x be given.
Assume H2.
Apply H1 to the current goal.
We use x to witness the existential quantifier.
An exact proof term for the current goal is H2.
Theorem. (iffEL)
∀A B : prop, (A B)AB
Proof:
An exact proof term for the current goal is (λA B ⇒ andEL (AB) (BA)).
Theorem. (iffER)
∀A B : prop, (A B)BA
Proof:
An exact proof term for the current goal is (λA B ⇒ andER (AB) (BA)).
Theorem. (iff_refl)
∀A : prop, A A
Proof:
An exact proof term for the current goal is (λA : propandI (AA) (AA) (λH : AH) (λH : AH)).
Theorem. (iff_sym)
∀A B : prop, (A B)(B A)
Proof:
Let A and B be given.
Assume H1: (AB) (BA).
Apply H1 to the current goal.
Assume H2: AB.
Assume H3: BA.
An exact proof term for the current goal is iffI B A H3 H2.
Theorem. (iff_trans)
∀A B C : prop, (A B)(B C)(A C)
Proof:
Let A, B and C be given.
Assume H1: A B.
Assume H2: B C.
Apply H1 to the current goal.
Assume H3: AB.
Assume H4: BA.
Apply H2 to the current goal.
Assume H5: BC.
Assume H6: CB.
An exact proof term for the current goal is (iffI A C (λH ⇒ H5 (H3 H)) (λH ⇒ H4 (H6 H))).
Theorem. (eq_i_tra)
∀x y z, x = yy = zx = z
Proof:
Let x, y and z be given.
Assume H1 H2.
rewrite the current goal using H2 (from right to left).
An exact proof term for the current goal is H1.
Theorem. (neq_i_sym)
∀x y, x yy x
Proof:
Let x and y be given.
Assume H1 H2.
Apply H1 to the current goal.
Use symmetry.
An exact proof term for the current goal is H2.
Theorem. (Eps_i_ex)
∀P : setprop, (∃x, P x)P (Eps_i P)
Proof:
Let P be given.
Assume H1.
Apply H1 to the current goal.
Let x be given.
Assume H2.
An exact proof term for the current goal is Eps_i_ax P x H2.
Theorem. (prop_ext_2)
∀p q : prop, (pq)(qp)p = q
Proof:
Let p and q be given.
Assume H1 H2.
Apply prop_ext to the current goal.
Apply iffI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
Theorem. (Subq_ref)
∀X : set, X X
Proof:
An exact proof term for the current goal is (λ(X x : set)(H : x X) ⇒ H).
Theorem. (Subq_tra)
∀X Y Z : set, X YY ZX Z
Proof:
An exact proof term for the current goal is (λ(X Y Z : set)(H1 : X Y)(H2 : Y Z)(x : set)(H : x X) ⇒ (H2 x (H1 x H))).
Theorem. (Empty_Subq_eq)
∀X : set, X EmptyX = Empty
Proof:
Let X be given.
Assume H1: X Empty.
Apply set_ext to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is (Subq_Empty X).
Theorem. (Empty_eq)
∀X : set, (∀x, x X)X = Empty
Proof:
Let X be given.
Assume H1: ∀x, x X.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume H2: x X.
We will prove False.
An exact proof term for the current goal is (H1 x H2).
Theorem. (UnionI)
∀X x Y : set, x YY Xx X
Proof:
Let X, x and Y be given.
Assume H1: x Y.
Assume H2: Y X.
Apply UnionEq X x to the current goal.
Assume _ H3.
Apply H3 to the current goal.
We will prove ∃Y : set, x Y Y X.
We use Y to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
Theorem. (UnionE)
∀X x : set, x X∃Y : set, x Y Y X
Proof:
An exact proof term for the current goal is (λX x : setiffEL (x X) (∃Y : set, x Y Y X) (UnionEq X x)).
Theorem. (UnionE_impred)
∀X x : set, x X∀p : prop, (∀Y : set, x YY Xp)p
Proof:
Let X and x be given.
Assume H1.
Let p be given.
Assume Hp.
Apply UnionE X x H1 to the current goal.
Let x be given.
Assume H2.
Apply H2 to the current goal.
An exact proof term for the current goal is Hp x.
Theorem. (PowerE)
∀X Y : set, Y 𝒫 XY X
Proof:
Let X and Y be given.
Apply PowerEq X Y to the current goal.
An exact proof term for the current goal is (λH _ ⇒ H).
Theorem. (Self_In_Power)
∀X : set, X 𝒫 X
Proof:
An exact proof term for the current goal is (λX : setPowerI X X (Subq_ref X)).
Theorem. (dneg)
∀P : prop, ¬ ¬ PP
Proof:
Let P be given.
Assume H1.
Apply xm P to the current goal.
An exact proof term for the current goal is (λH ⇒ H).
Assume H2: ¬ P.
We will prove False.
An exact proof term for the current goal is H1 H2.
Theorem. (not_all_ex_demorgan_i)
∀P : setprop, ¬ (∀x, P x)∃x, ¬ P x
Proof:
Let P be given.
Assume u: ¬ ∀x, P x.
Apply dneg to the current goal.
Assume v: ¬ ∃x, ¬ P x.
Apply u to the current goal.
Let x be given.
Apply dneg to the current goal.
Assume w: ¬ P x.
An exact proof term for the current goal is (not_ex_all_demorgan_i (λx ⇒ ¬ P x) v x w).
Theorem. (eq_or_nand)
or = (λx y : prop¬ (¬ x ¬ y))
Proof:
Apply func_ext prop (propprop) to the current goal.
Let x be given.
Apply func_ext prop prop to the current goal.
Let y be given.
Apply prop_ext_2 to the current goal.
Assume H1: x y.
Assume H2: ¬ x ¬ y.
Apply H2 to the current goal.
Assume H3 H4.
An exact proof term for the current goal is (H1 False H3 H4).
Assume H1: ¬ (¬ x ¬ y).
Apply (xm x) to the current goal.
Assume H2: x.
Apply orIL to the current goal.
An exact proof term for the current goal is H2.
Assume H2: ¬ x.
Apply (xm y) to the current goal.
Assume H3: y.
Apply orIR to the current goal.
An exact proof term for the current goal is H3.
Assume H3: ¬ y.
Apply H1 to the current goal.
An exact proof term for the current goal is (andI (¬ x) (¬ y) H2 H3).
Definition. We define exactly1of2 to be λA B : propA ¬ B ¬ A B of type proppropprop.
Theorem. (exactly1of2_I1)
∀A B : prop, A¬ Bexactly1of2 A B
Proof:
Let A and B be given.
Assume HA: A.
Assume HB: ¬ B.
We will prove A ¬ B ¬ A B.
Apply orIL to the current goal.
We will prove A ¬ B.
An exact proof term for the current goal is (andI A (¬ B) HA HB).
Theorem. (exactly1of2_I2)
∀A B : prop, ¬ ABexactly1of2 A B
Proof:
Let A and B be given.
Assume HA: ¬ A.
Assume HB: B.
We will prove A ¬ B ¬ A B.
Apply orIR to the current goal.
We will prove ¬ A B.
An exact proof term for the current goal is (andI (¬ A) B HA HB).
Theorem. (exactly1of2_E)
∀A B : prop, exactly1of2 A B∀p : prop, (A¬ Bp)(¬ ABp)p
Proof:
Let A and B be given.
Assume H1: exactly1of2 A B.
Let p be given.
Assume H2: A¬ Bp.
Assume H3: ¬ ABp.
Apply (H1 p) to the current goal.
An exact proof term for the current goal is (λH4 : A ¬ BH4 p H2).
An exact proof term for the current goal is (λH4 : ¬ A BH4 p H3).
Theorem. (exactly1of2_or)
∀A B : prop, exactly1of2 A BA B
Proof:
Let A and B be given.
Assume H1: exactly1of2 A B.
Apply (exactly1of2_E A B H1 (A B)) to the current goal.
An exact proof term for the current goal is (λ(HA : A)(_ : ¬ B) ⇒ orIL A B HA).
An exact proof term for the current goal is (λ(_ : ¬ A)(HB : B) ⇒ orIR A B HB).
Theorem. (ReplI)
∀A : set, ∀F : setset, ∀x : set, x AF x {F x|xA}
Proof:
Let A, F and x be given.
Assume H1.
Apply ReplEq A F (F x) to the current goal.
Assume _ H2.
Apply H2 to the current goal.
We will prove ∃x'A, F x = F x'.
We use x to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is (λq H ⇒ H).
Theorem. (ReplE)
∀A : set, ∀F : setset, ∀y : set, y {F x|xA}∃xA, y = F x
Proof:
Let A, F and y be given.
Apply ReplEq A F y to the current goal.
An exact proof term for the current goal is (λH _ ⇒ H).
Theorem. (ReplE_impred)
∀A : set, ∀F : setset, ∀y : set, y {F x|xA}∀p : prop, (∀x : set, x Ay = F xp)p
Proof:
Let A, F and y be given.
Assume H1.
Apply ReplE A F y H1 to the current goal.
Let x be given.
Assume H2.
Apply H2 to the current goal.
Assume H3 H4.
Let p be given.
Assume Hp.
An exact proof term for the current goal is Hp x H3 H4.
Theorem. (ReplE')
∀X, ∀f : setset, ∀p : setprop, (∀xX, p (f x))∀y{f x|xX}, p y
Proof:
Let X, f and p be given.
Assume H1.
Let y be given.
Assume Hy.
Apply ReplE_impred X f y Hy to the current goal.
Let x be given.
Assume Hx: x X.
Assume Hx2: y = f x.
We will prove p y.
rewrite the current goal using Hx2 (from left to right).
An exact proof term for the current goal is H1 x Hx.
Theorem. (Repl_Empty)
∀F : setset, {F x|xEmpty} = Empty
Proof:
Let F be given.
Apply (Empty_eq {F x|xEmpty}) to the current goal.
Let y be given.
Assume H1: y {F x|xEmpty}.
Apply (ReplE_impred Empty F y H1) to the current goal.
Let x be given.
Assume H2: x Empty.
Assume _.
An exact proof term for the current goal is (EmptyE x H2).
Theorem. (ReplEq_ext_sub)
∀X, ∀F G : setset, (∀xX, F x = G x){F x|xX} {G x|xX}
Proof:
Let X, F and G be given.
Assume H1: ∀xX, F x = G x.
Let y be given.
Assume Hy: y {F x|xX}.
Apply ReplE_impred X F y Hy to the current goal.
Let x be given.
Assume Hx: x X.
Assume H2: y = F x.
We will prove y {G x|xX}.
rewrite the current goal using H2 (from left to right).
We will prove F x {G x|xX}.
rewrite the current goal using H1 x Hx (from left to right).
We will prove G x {G x|xX}.
Apply ReplI to the current goal.
An exact proof term for the current goal is Hx.
Theorem. (ReplEq_ext)
∀X, ∀F G : setset, (∀xX, F x = G x){F x|xX} = {G x|xX}
Proof:
Let X, F and G be given.
Assume H1: ∀xX, F x = G x.
Apply set_ext to the current goal.
An exact proof term for the current goal is ReplEq_ext_sub X F G H1.
Apply ReplEq_ext_sub X G F to the current goal.
Let x be given.
Assume Hx.
Use symmetry.
An exact proof term for the current goal is H1 x Hx.
Theorem. (Repl_inv_eq)
∀P : setprop, ∀f g : setset, (∀x, P xg (f x) = x)∀X, (∀xX, P x){g y|y{f x|xX}} = X
Proof:
Let P, f and g be given.
Assume H1.
Let X be given.
Assume HX.
Apply set_ext to the current goal.
Let w be given.
Assume Hw: w {g y|y{f x|xX}}.
Apply ReplE_impred {f x|xX} g w Hw to the current goal.
Let y be given.
Assume Hy: y {f x|xX}.
Assume Hwy: w = g y.
Apply ReplE_impred X f y Hy to the current goal.
Let x be given.
Assume Hx: x X.
Assume Hyx: y = f x.
We will prove w X.
rewrite the current goal using Hwy (from left to right).
rewrite the current goal using Hyx (from left to right).
We will prove g (f x) X.
rewrite the current goal using H1 x (HX x Hx) (from left to right).
An exact proof term for the current goal is Hx.
Let x be given.
Assume Hx: x X.
rewrite the current goal using H1 x (HX x Hx) (from right to left).
We will prove g (f x) {g y|y{f x|xX}}.
Apply ReplI to the current goal.
We will prove f x {f x|xX}.
Apply ReplI to the current goal.
An exact proof term for the current goal is Hx.
Theorem. (Repl_invol_eq)
∀P : setprop, ∀f : setset, (∀x, P xf (f x) = x)∀X, (∀xX, P x){f y|y{f x|xX}} = X
Proof:
Let P and f be given.
Assume H1.
An exact proof term for the current goal is Repl_inv_eq P f f H1.
Definition. We define If_i to be (λp x y ⇒ Eps_i (λz : setp z = x ¬ p z = y)) of type propsetsetset.
Notation. if cond then T else E is notation corresponding to If_i type cond T E where type is the inferred type of T.
Theorem. (If_i_correct)
∀p : prop, ∀x y : set, p (if p then x else y) = x ¬ p (if p then x else y) = y
Proof:
Let p, x and y be given.
Apply (xm p) to the current goal.
Assume H1: p.
We prove the intermediate claim L1: p x = x ¬ p x = y.
Apply orIL to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Use reflexivity.
An exact proof term for the current goal is (Eps_i_ax (λz : setp z = x ¬ p z = y) x L1).
Assume H1: ¬ p.
We prove the intermediate claim L1: p y = x ¬ p y = y.
Apply orIR to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Use reflexivity.
An exact proof term for the current goal is (Eps_i_ax (λz : setp z = x ¬ p z = y) y L1).
Theorem. (If_i_0)
∀p : prop, ∀x y : set, ¬ p(if p then x else y) = y
Proof:
Let p, x and y be given.
Assume H1: ¬ p.
Apply (If_i_correct p x y) to the current goal.
Assume H2: p (if p then x else y) = x.
An exact proof term for the current goal is (H1 (andEL p ((if p then x else y) = x) H2) ((if p then x else y) = y)).
Assume H2: ¬ p (if p then x else y) = y.
An exact proof term for the current goal is (andER (¬ p) ((if p then x else y) = y) H2).
Theorem. (If_i_1)
∀p : prop, ∀x y : set, p(if p then x else y) = x
Proof:
Let p, x and y be given.
Assume H1: p.
Apply (If_i_correct p x y) to the current goal.
Assume H2: p (if p then x else y) = x.
An exact proof term for the current goal is (andER p ((if p then x else y) = x) H2).
Assume H2: ¬ p (if p then x else y) = y.
An exact proof term for the current goal is (andEL (¬ p) ((if p then x else y) = y) H2 H1 ((if p then x else y) = x)).
Theorem. (If_i_or)
∀p : prop, ∀x y : set, (if p then x else y) = x (if p then x else y) = y
Proof:
Let p, x and y be given.
Apply (xm p) to the current goal.
Assume H1: p.
Apply orIL to the current goal.
An exact proof term for the current goal is (If_i_1 p x y H1).
Assume H1: ¬ p.
Apply orIR to the current goal.
An exact proof term for the current goal is (If_i_0 p x y H1).
Definition. We define UPair to be λy z ⇒ {if Empty X then y else z|X𝒫 (𝒫 Empty)} of type setsetset.
Notation. {x,y} is notation for UPair x y.
Theorem. (UPairE)
∀x y z : set, x {y,z}x = y x = z
Proof:
Let x, y and z be given.
Assume H1: x {y,z}.
Apply (ReplE (𝒫 (𝒫 Empty)) (λX ⇒ if Empty X then y else z) x H1) to the current goal.
Let X be given.
Assume H2: X 𝒫 (𝒫 Empty) x = if Empty X then y else z.
We prove the intermediate claim L1: x = if Empty X then y else z.
An exact proof term for the current goal is (andER (X 𝒫 (𝒫 Empty)) (x = if Empty X then y else z) H2).
Apply (If_i_or (Empty X) y z) to the current goal.
Assume H3: (if Empty X then y else z) = y.
Apply orIL to the current goal.
We will prove x = y.
rewrite the current goal using H3 (from right to left).
An exact proof term for the current goal is L1.
Assume H3: (if Empty X then y else z) = z.
Apply orIR to the current goal.
We will prove x = z.
rewrite the current goal using H3 (from right to left).
An exact proof term for the current goal is L1.
Theorem. (UPairI1)
∀y z : set, y {y,z}
Proof:
Let y and z be given.
We will prove y {y,z}.
rewrite the current goal using (If_i_1 (Empty 𝒫 Empty) y z (Empty_In_Power Empty)) (from right to left) at position 1.
We will prove (if Empty 𝒫 Empty then y else z) {y,z}.
Apply (ReplI (𝒫 (𝒫 Empty)) (λX : setif (Empty X) then y else z) (𝒫 Empty)) to the current goal.
We will prove 𝒫 Empty 𝒫 (𝒫 Empty).
An exact proof term for the current goal is (Self_In_Power (𝒫 Empty)).
Theorem. (UPairI2)
∀y z : set, z {y,z}
Proof:
Let y and z be given.
We will prove z {y,z}.
rewrite the current goal using (If_i_0 (Empty Empty) y z (EmptyE Empty)) (from right to left) at position 1.
We will prove (if Empty Empty then y else z) {y,z}.
Apply (ReplI (𝒫 (𝒫 Empty)) (λX : setif (Empty X) then y else z) Empty) to the current goal.
We will prove Empty 𝒫 (𝒫 Empty).
An exact proof term for the current goal is (Empty_In_Power (𝒫 Empty)).
Definition. We define Sing to be λx ⇒ {x,x} of type setset.
Notation. {x} is notation for Sing x.
Theorem. (SingI)
∀x : set, x {x}
Proof:
An exact proof term for the current goal is (λx : setUPairI1 x x).
Theorem. (SingE)
∀x y : set, y {x}y = x
Proof:
An exact proof term for the current goal is (λx y H ⇒ UPairE y x x H (y = x) (λH ⇒ H) (λH ⇒ H)).
Definition. We define binunion to be λX Y ⇒ {X,Y} of type setsetset.
Notation. We use as an infix operator with priority 345 and which associates to the left corresponding to applying term binunion.
Theorem. (binunionI1)
∀X Y z : set, z Xz X Y
Proof:
Let X, Y and z be given.
Assume H1: z X.
We will prove z X Y.
We will prove z {X,Y}.
Apply (UnionI {X,Y} z X) to the current goal.
We will prove z X.
An exact proof term for the current goal is H1.
We will prove X {X,Y}.
An exact proof term for the current goal is (UPairI1 X Y).
Theorem. (binunionI2)
∀X Y z : set, z Yz X Y
Proof:
Let X, Y and z be given.
Assume H1: z Y.
We will prove z X Y.
We will prove z {X,Y}.
Apply (UnionI {X,Y} z Y) to the current goal.
We will prove z Y.
An exact proof term for the current goal is H1.
We will prove Y {X,Y}.
An exact proof term for the current goal is (UPairI2 X Y).
Theorem. (binunionE)
∀X Y z : set, z X Yz X z Y
Proof:
Let X, Y and z be given.
Assume H1: z X Y.
We will prove z X z Y.
Apply (UnionE_impred {X,Y} z H1) to the current goal.
Let Z be given.
Assume H2: z Z.
Assume H3: Z {X,Y}.
Apply (UPairE Z X Y H3) to the current goal.
Assume H4: Z = X.
Apply orIL to the current goal.
We will prove z X.
rewrite the current goal using H4 (from right to left).
We will prove z Z.
An exact proof term for the current goal is H2.
Assume H4: Z = Y.
Apply orIR to the current goal.
We will prove z Y.
rewrite the current goal using H4 (from right to left).
We will prove z Z.
An exact proof term for the current goal is H2.
Theorem. (binunionE')
∀X Y z, ∀p : prop, (z Xp)(z Yp)(z X Yp)
Proof:
Let X, Y, z and p be given.
Assume H1 H2 Hz.
Apply binunionE X Y z Hz to the current goal.
Assume H3: z X.
An exact proof term for the current goal is H1 H3.
Assume H3: z Y.
An exact proof term for the current goal is H2 H3.
Theorem. (binunion_asso)
∀X Y Z : set, X (Y Z) = (X Y) Z
Proof:
Let X, Y and Z be given.
Apply set_ext to the current goal.
Let w be given.
Assume H1: w X (Y Z).
We will prove w (X Y) Z.
Apply (binunionE X (Y Z) w H1) to the current goal.
Assume H2: w X.
Apply binunionI1 to the current goal.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is H2.
Assume H2: w Y Z.
Apply (binunionE Y Z w H2) to the current goal.
Assume H3: w Y.
Apply binunionI1 to the current goal.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H3.
Assume H3: w Z.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H3.
Let w be given.
Assume H1: w (X Y) Z.
We will prove w X (Y Z).
Apply (binunionE (X Y) Z w H1) to the current goal.
Assume H2: w X Y.
Apply (binunionE X Y w H2) to the current goal.
Assume H3: w X.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is H3.
Assume H3: w Y.
Apply binunionI2 to the current goal.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is H3.
Assume H2: w Z.
Apply binunionI2 to the current goal.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H2.
Theorem. (binunion_com_Subq)
∀X Y : set, X Y Y X
Proof:
Let X, Y and w be given.
Assume H1: w X Y.
We will prove w Y X.
Apply (binunionE X Y w H1) to the current goal.
Assume H2: w X.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H2.
Assume H2: w Y.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is H2.
Theorem. (binunion_com)
∀X Y : set, X Y = Y X
Proof:
Let X and Y be given.
Apply set_ext to the current goal.
An exact proof term for the current goal is (binunion_com_Subq X Y).
An exact proof term for the current goal is (binunion_com_Subq Y X).
Theorem. (binunion_idl)
∀X : set, Empty X = X
Proof:
Let X be given.
Apply set_ext to the current goal.
Let x be given.
Assume H1: x Empty X.
Apply (binunionE Empty X x H1) to the current goal.
Assume H2: x Empty.
We will prove False.
An exact proof term for the current goal is (EmptyE x H2).
Assume H2: x X.
An exact proof term for the current goal is H2.
Let x be given.
Assume H2: x X.
We will prove x Empty X.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H2.
Theorem. (binunion_idr)
∀X : set, X Empty = X
Proof:
Let X be given.
rewrite the current goal using (binunion_com X Empty) (from left to right).
An exact proof term for the current goal is (binunion_idl X).
Theorem. (binunion_Subq_1)
∀X Y : set, X X Y
Proof:
An exact proof term for the current goal is binunionI1.
Theorem. (binunion_Subq_2)
∀X Y : set, Y X Y
Proof:
An exact proof term for the current goal is binunionI2.
Theorem. (binunion_Subq_min)
∀X Y Z : set, X ZY ZX Y Z
Proof:
Let X, Y and Z be given.
Assume H1: X Z.
Assume H2: Y Z.
Let w be given.
Assume H3: w X Y.
Apply (binunionE X Y w H3) to the current goal.
Assume H4: w X.
An exact proof term for the current goal is (H1 w H4).
Assume H4: w Y.
An exact proof term for the current goal is (H2 w H4).
Theorem. (Subq_binunion_eq)
∀X Y, (X Y) = (X Y = Y)
Proof:
Let X and Y be given.
Apply prop_ext_2 to the current goal.
Assume H1: X Y.
We will prove X Y = Y.
Apply set_ext to the current goal.
We will prove X Y Y.
Apply (binunion_Subq_min X Y Y) to the current goal.
We will prove X Y.
An exact proof term for the current goal is H1.
We will prove Y Y.
An exact proof term for the current goal is (Subq_ref Y).
We will prove Y X Y.
An exact proof term for the current goal is (binunion_Subq_2 X Y).
Assume H1: X Y = Y.
We will prove X Y.
rewrite the current goal using H1 (from right to left).
We will prove X X Y.
An exact proof term for the current goal is (binunion_Subq_1 X Y).
Definition. We define SetAdjoin to be λX y ⇒ X {y} of type setsetset.
Notation. We now use the set enumeration notation {...,...,...} in general. If 0 elements are given, then Empty is used to form the corresponding term. If 1 element is given, then Sing is used to form the corresponding term. If 2 elements are given, then UPair is used to form the corresponding term. If more than elements are given, then SetAdjoin is used to reduce to the case with one fewer elements.
Definition. We define famunion to be λX F ⇒ {F x|xX} of type set(setset)set.
Notation. We use x [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using famunion.
Theorem. (famunionI)
∀X : set, ∀F : (setset), ∀x y : set, x Xy F xy xXF x
Proof:
An exact proof term for the current goal is (λX F x y H1 H2 ⇒ UnionI (Repl X F) y (F x) H2 (ReplI X F x H1)).
Theorem. (famunionE)
∀X : set, ∀F : (setset), ∀y : set, y (xXF x)∃xX, y F x
Proof:
Let X, F and y be given.
Assume H1: y (xXF x).
We will prove ∃xX, y F x.
Apply (UnionE_impred {F x|xX} y H1) to the current goal.
Let Y be given.
Assume H2: y Y.
Assume H3: Y {F x|xX}.
Apply (ReplE_impred X F Y H3) to the current goal.
Let x be given.
Assume H4: x X.
Assume H5: Y = F x.
We use x to witness the existential quantifier.
We will prove x X y F x.
Apply andI to the current goal.
An exact proof term for the current goal is H4.
We will prove y F x.
rewrite the current goal using H5 (from right to left).
An exact proof term for the current goal is H2.
Theorem. (famunionE_impred)
∀X : set, ∀F : (setset), ∀y : set, y (xXF x)∀p : prop, (∀x, x Xy F xp)p
Proof:
Let X, F and y be given.
Assume Hy.
Let p be given.
Assume Hp.
Apply famunionE X F y Hy to the current goal.
Let x be given.
Assume H1.
Apply H1 to the current goal.
An exact proof term for the current goal is Hp x.
Theorem. (famunion_Empty)
∀F : setset, (xEmptyF x) = Empty
Proof:
Let F be given.
Apply Empty_Subq_eq to the current goal.
Let y be given.
Assume Hy: y xEmptyF x.
Apply famunionE_impred Empty F y Hy to the current goal.
Let x be given.
Assume Hx: x Empty.
We will prove False.
An exact proof term for the current goal is EmptyE x Hx.
Theorem. (famunion_Subq)
∀X, ∀f g : setset, (∀xX, f x g x)famunion X f famunion X g
Proof:
Let X, f and g be given.
Assume Hfg.
Let y be given.
Assume Hy.
Apply famunionE_impred X f y Hy to the current goal.
Let x be given.
Assume Hx.
Assume H1: y f x.
Apply famunionI X g x y Hx to the current goal.
We will prove y g x.
An exact proof term for the current goal is Hfg x Hx y H1.
Theorem. (famunion_ext)
∀X, ∀f g : setset, (∀xX, f x = g x)famunion X f = famunion X g
Proof:
Let X, f and g be given.
Assume Hfg.
Apply set_ext to the current goal.
Apply famunion_Subq to the current goal.
Let x be given.
Assume Hx.
rewrite the current goal using Hfg x Hx (from left to right).
Apply Subq_ref to the current goal.
Apply famunion_Subq to the current goal.
Let x be given.
Assume Hx.
rewrite the current goal using Hfg x Hx (from left to right).
Apply Subq_ref to the current goal.
Beginning of Section SepSec
Variable X : set
Variable P : setprop
Let z : setEps_i (λz ⇒ z X P z)
Let F : setsetλx ⇒ if P x then x else z
Definition. We define Sep to be if (∃zX, P z) then {F x|xX} else Empty of type set.
End of Section SepSec
Notation. {xA | B} is notation for Sep Ax . B).
Theorem. (SepI)
∀X : set, ∀P : (setprop), ∀x : set, x XP xx {xX|P x}
Proof:
Let X, P and x be given.
Set z to be the term Eps_i (λz ⇒ z X P z) of type set.
Set F to be the term λx ⇒ if P x then x else z of type setset.
Assume H1: x X.
Assume H2: P x.
We prove the intermediate claim L1: ∃zX, P z.
We use x to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
We will prove x {xX|P x}.
We will prove x if (∃zX, P z) then {F x|xX} else Empty.
We prove the intermediate claim L2: (if (∃zX, P z) then {F x|xX} else Empty) = {F x|xX}.
An exact proof term for the current goal is (If_i_1 (∃zX, P z) {F x|xX} Empty L1).
rewrite the current goal using L2 (from left to right).
We will prove x {F x|xX}.
We prove the intermediate claim L3: F x = x.
We will prove (if P x then x else z) = x.
An exact proof term for the current goal is (If_i_1 (P x) x z H2).
rewrite the current goal using L3 (from right to left).
We will prove F x {F x|xX}.
An exact proof term for the current goal is (ReplI X F x H1).
Theorem. (SepE)
∀X : set, ∀P : (setprop), ∀x : set, x {xX|P x}x X P x
Proof:
Let X, P and x be given.
Set z to be the term Eps_i (λz ⇒ z X P z) of type set.
Set F to be the term λx ⇒ if P x then x else z of type setset.
Apply (xm (∃zX, P z)) to the current goal.
Assume H1: ∃zX, P z.
We will prove (x (if (∃zX, P z) then {F x|xX} else Empty)x X P x).
We prove the intermediate claim L1: (if (∃zX, P z) then {F x|xX} else Empty) = {F x|xX}.
An exact proof term for the current goal is (If_i_1 (∃zX, P z) {F x|xX} Empty H1).
rewrite the current goal using L1 (from left to right).
We will prove x {F x|xX}x X P x.
Assume H2: x {F x|xX}.
Apply (ReplE_impred X F x H2) to the current goal.
Let y be given.
Assume H3: y X.
Assume H4: x = F y.
We will prove x X P x.
Apply (xm (P y)) to the current goal.
Assume H5: P y.
We prove the intermediate claim L2: x = y.
rewrite the current goal using (If_i_1 (P y) y z H5) (from right to left).
An exact proof term for the current goal is H4.
rewrite the current goal using L2 (from left to right).
We will prove y X P y.
Apply andI to the current goal.
An exact proof term for the current goal is H3.
An exact proof term for the current goal is H5.
Assume H5: ¬ P y.
We prove the intermediate claim L2: x = z.
rewrite the current goal using (If_i_0 (P y) y z H5) (from right to left).
An exact proof term for the current goal is H4.
rewrite the current goal using L2 (from left to right).
We will prove z X P z.
An exact proof term for the current goal is (Eps_i_ex (λz ⇒ z X P z) H1).
Assume H1: ¬ ∃zX, P z.
We will prove (x (if (∃zX, P z) then {F x|xX} else Empty)x X P x).
We prove the intermediate claim L1: (if (∃zX, P z) then {F x|xX} else Empty) = Empty.
An exact proof term for the current goal is (If_i_0 (∃zX, P z) {F x|xX} Empty H1).
rewrite the current goal using L1 (from left to right).
We will prove x Emptyx X P x.
Assume H2: x Empty.
We will prove False.
An exact proof term for the current goal is (EmptyE x H2).
Theorem. (SepE1)
∀X : set, ∀P : (setprop), ∀x : set, x {xX|P x}x X
Proof:
An exact proof term for the current goal is (λX P x H ⇒ SepE X P x H (x X) (λH _ ⇒ H)).
Theorem. (SepE2)
∀X : set, ∀P : (setprop), ∀x : set, x {xX|P x}P x
Proof:
An exact proof term for the current goal is (λX P x H ⇒ SepE X P x H (P x) (λ_ H ⇒ H)).
Theorem. (Sep_Empty)
∀P : setprop, {xEmpty|P x} = Empty
Proof:
Let P be given.
Apply Empty_eq to the current goal.
Let x be given.
Assume Hx.
An exact proof term for the current goal is EmptyE x (SepE1 Empty P x Hx).
Theorem. (Sep_Subq)
∀X : set, ∀P : setprop, {xX|P x} X
Proof:
An exact proof term for the current goal is SepE1.
Theorem. (Sep_In_Power)
∀X : set, ∀P : setprop, {xX|P x} 𝒫 X
Proof:
An exact proof term for the current goal is (λX P ⇒ PowerI X (Sep X P) (Sep_Subq X P)).
Definition. We define ReplSep to be λX P F ⇒ {F x|x{zX|P z}} of type set(setprop)(setset)set.
Notation. {B| xA, C} is notation for ReplSep Ax . C) (λ x . B).
Theorem. (ReplSepI)
∀X : set, ∀P : setprop, ∀F : setset, ∀x : set, x XP xF x {F x|xX, P x}
Proof:
An exact proof term for the current goal is (λX P F x u v ⇒ ReplI (Sep X P) F x (SepI X P x u v)).
Theorem. (ReplSepE)
∀X : set, ∀P : setprop, ∀F : setset, ∀y : set, y {F x|xX, P x}∃x : set, x X P x y = F x
Proof:
Let X, P, F and y be given.
Assume H1: y {F x|x{zX|P z}}.
Apply (ReplE {zX|P z} F y H1) to the current goal.
Let x be given.
Assume H2: x {zX|P z} y = F x.
Apply H2 to the current goal.
Assume H3: x {zX|P z}.
Assume H4: y = F x.
Apply (SepE X P x H3) to the current goal.
Assume H5: x X.
Assume H6: P x.
We use x to witness the existential quantifier.
Apply and3I to the current goal.
An exact proof term for the current goal is H5.
An exact proof term for the current goal is H6.
An exact proof term for the current goal is H4.
Theorem. (ReplSepE_impred)
∀X : set, ∀P : setprop, ∀F : setset, ∀y : set, y {F x|xX, P x}∀p : prop, (∀xX, P xy = F xp)p
Proof:
Let X, P, F and y be given.
Assume H1: y {F x|xX, P x}.
Let p be given.
Assume H2: ∀xX, P xy = F xp.
We will prove p.
Apply ReplSepE X P F y H1 to the current goal.
Let x be given.
Assume H3.
Apply H3 to the current goal.
Assume H3.
Apply H3 to the current goal.
An exact proof term for the current goal is H2 x.
Definition. We define binintersect to be λX Y ⇒ {xX|x Y} of type setsetset.
Notation. We use as an infix operator with priority 340 and which associates to the left corresponding to applying term binintersect.
Theorem. (binintersectI)
∀X Y z, z Xz Yz X Y
Proof:
An exact proof term for the current goal is (λX Y z H1 H2 ⇒ SepI X (λx : setx Y) z H1 H2).
Theorem. (binintersectE)
∀X Y z, z X Yz X z Y
Proof:
An exact proof term for the current goal is (λX Y z H1 ⇒ SepE X (λx : setx Y) z H1).
Theorem. (binintersectE1)
∀X Y z, z X Yz X
Proof:
An exact proof term for the current goal is (λX Y z H1 ⇒ SepE1 X (λx : setx Y) z H1).
Theorem. (binintersectE2)
∀X Y z, z X Yz Y
Proof:
An exact proof term for the current goal is (λX Y z H1 ⇒ SepE2 X (λx : setx Y) z H1).
Theorem. (binintersect_Subq_1)
∀X Y : set, X Y X
Proof:
An exact proof term for the current goal is binintersectE1.
Theorem. (binintersect_Subq_2)
∀X Y : set, X Y Y
Proof:
An exact proof term for the current goal is binintersectE2.
Theorem. (binintersect_Subq_eq_1)
∀X Y, X YX Y = X
Proof:
Let X and Y be given.
Assume H1: X Y.
Apply set_ext to the current goal.
Apply binintersect_Subq_1 to the current goal.
Let x be given.
Assume H2: x X.
Apply binintersectI to the current goal.
An exact proof term for the current goal is H2.
Apply H1 to the current goal.
An exact proof term for the current goal is H2.
Theorem. (binintersect_Subq_max)
∀X Y Z : set, Z XZ YZ X Y
Proof:
Let X, Y and Z be given.
Assume H1: Z X.
Assume H2: Z Y.
Let w be given.
Assume H3: w Z.
Apply (binintersectI X Y w) to the current goal.
We will prove w X.
An exact proof term for the current goal is (H1 w H3).
We will prove w Y.
An exact proof term for the current goal is (H2 w H3).
Theorem. (binintersect_com_Subq)
∀X Y : set, X Y Y X
Proof:
Let X and Y be given.
Apply (binintersect_Subq_max Y X (X Y)) to the current goal.
We will prove X Y Y.
Apply binintersect_Subq_2 to the current goal.
We will prove X Y X.
Apply binintersect_Subq_1 to the current goal.
Theorem. (binintersect_com)
∀X Y : set, X Y = Y X
Proof:
Let X and Y be given.
Apply set_ext to the current goal.
An exact proof term for the current goal is (binintersect_com_Subq X Y).
An exact proof term for the current goal is (binintersect_com_Subq Y X).
Definition. We define setminus to be λX Y ⇒ Sep X (λx ⇒ x Y) of type setsetset.
Notation. We use as an infix operator with priority 350 and no associativity corresponding to applying term setminus.
Theorem. (setminusI)
∀X Y z, (z X)(z Y)z X Y
Proof:
An exact proof term for the current goal is (λX Y z H1 H2 ⇒ SepI X (λx : setx Y) z H1 H2).
Theorem. (setminusE)
∀X Y z, (z X Y)z X z Y
Proof:
An exact proof term for the current goal is (λX Y z H ⇒ SepE X (λx : setx Y) z H).
Theorem. (setminusE1)
∀X Y z, (z X Y)z X
Proof:
An exact proof term for the current goal is (λX Y z H ⇒ SepE1 X (λx : setx Y) z H).
Theorem. (setminusE2)
∀X Y z, (z X Y)z Y
Proof:
An exact proof term for the current goal is (λX Y z H ⇒ SepE2 X (λx : setx Y) z H).
Theorem. (setminus_Subq)
∀X Y : set, X Y X
Proof:
An exact proof term for the current goal is setminusE1.
Theorem. (setminus_In_Power)
∀A U, A U 𝒫 A
Proof:
Let A and U be given.
Apply PowerI to the current goal.
Apply setminus_Subq to the current goal.
Theorem. (binunion_remove1_eq)
∀X, ∀xX, X = (X {x}) {x}
Proof:
Let X and x be given.
Assume Hx: x X.
Apply set_ext to the current goal.
Let y be given.
Assume Hy: y X.
We will prove y (X {x}) {x}.
Apply xm (y {x}) to the current goal.
Assume H1: y {x}.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H1.
Assume H1: y {x}.
Apply binunionI1 to the current goal.
Apply setminusI to the current goal.
An exact proof term for the current goal is Hy.
An exact proof term for the current goal is H1.
Let y be given.
Assume Hy: y (X {x}) {x}.
Apply binunionE (X {x}) {x} y Hy to the current goal.
Assume H1: y X {x}.
We will prove y X.
An exact proof term for the current goal is setminusE1 X {x} y H1.
Assume H1: y {x}.
We will prove y X.
rewrite the current goal using SingE x y H1 (from left to right).
We will prove x X.
An exact proof term for the current goal is Hx.
Theorem. (In_irref)
∀x, x x
Proof:
Apply In_ind to the current goal.
We will prove (∀X : set, (∀x : set, x Xx x)X X).
Let X be given.
Assume IH: ∀x : set, x Xx x.
Assume H: X X.
An exact proof term for the current goal is IH X H H.
Theorem. (In_no2cycle)
∀x y, x yy xFalse
Proof:
Apply In_ind to the current goal.
Let x be given.
Assume IH: ∀z, z x∀y, z yy zFalse.
Let y be given.
Assume H1: x y.
Assume H2: y x.
An exact proof term for the current goal is IH y H2 x H2 H1.
Definition. We define ordsucc to be λx : setx {x} of type setset.
Theorem. (ordsuccI1)
∀x : set, x ordsucc x
Proof:
Let x be given.
An exact proof term for the current goal is (λ(y : set)(H1 : y x) ⇒ binunionI1 x {x} y H1).
Theorem. (ordsuccI2)
∀x : set, x ordsucc x
Proof:
An exact proof term for the current goal is (λx : setbinunionI2 x {x} x (SingI x)).
Theorem. (ordsuccE)
∀x y : set, y ordsucc xy x y = x
Proof:
Let x and y be given.
Assume H1: y x {x}.
Apply (binunionE x {x} y H1) to the current goal.
Assume H2: y x.
Apply orIL to the current goal.
An exact proof term for the current goal is H2.
Assume H2: y {x}.
Apply orIR to the current goal.
An exact proof term for the current goal is (SingE x y H2).
Notation. Natural numbers 0,1,2,... are notation for the terms formed using Empty as 0 and forming successors with ordsucc.
Theorem. (neq_0_ordsucc)
∀a : set, 0 ordsucc a
Proof:
Let a be given.
We will prove ¬ (0 = ordsucc a).
Assume H1: 0 = ordsucc a.
We prove the intermediate claim L1: a ordsucc aFalse.
rewrite the current goal using H1 (from right to left).
An exact proof term for the current goal is (EmptyE a).
An exact proof term for the current goal is (L1 (ordsuccI2 a)).
Theorem. (neq_ordsucc_0)
∀a : set, ordsucc a 0
Proof:
Let a be given.
An exact proof term for the current goal is neq_i_sym 0 (ordsucc a) (neq_0_ordsucc a).
Theorem. (ordsucc_inj)
∀a b : set, ordsucc a = ordsucc ba = b
Proof:
Let a and b be given.
Assume H1: ordsucc a = ordsucc b.
We prove the intermediate claim L1: a ordsucc b.
rewrite the current goal using H1 (from right to left).
An exact proof term for the current goal is (ordsuccI2 a).
Apply (ordsuccE b a L1) to the current goal.
Assume H2: a b.
We prove the intermediate claim L2: b ordsucc a.
rewrite the current goal using H1 (from left to right).
An exact proof term for the current goal is (ordsuccI2 b).
Apply (ordsuccE a b L2) to the current goal.
Assume H3: b a.
We will prove False.
An exact proof term for the current goal is (In_no2cycle a b H2 H3).
Assume H3: b = a.
Use symmetry.
An exact proof term for the current goal is H3.
Assume H2: a = b.
An exact proof term for the current goal is H2.
Theorem. (In_0_1)
Proof:
An exact proof term for the current goal is (ordsuccI2 0).
Theorem. (In_0_2)
Proof:
An exact proof term for the current goal is (ordsuccI1 1 0 In_0_1).
Theorem. (In_1_2)
Proof:
An exact proof term for the current goal is (ordsuccI2 1).
Definition. We define nat_p to be λn : set∀p : setprop, p 0(∀x : set, p xp (ordsucc x))p n of type setprop.
Theorem. (nat_0)
Proof:
An exact proof term for the current goal is (λp H _ ⇒ H).
Theorem. (nat_ordsucc)
∀n : set, nat_p nnat_p (ordsucc n)
Proof:
An exact proof term for the current goal is (λn H1 p H2 H3 ⇒ H3 n (H1 p H2 H3)).
Theorem. (nat_1)
Proof:
An exact proof term for the current goal is (nat_ordsucc 0 nat_0).
Theorem. (nat_2)
Proof:
An exact proof term for the current goal is (nat_ordsucc 1 nat_1).
Theorem. (nat_0_in_ordsucc)
∀n, nat_p n0 ordsucc n
Proof:
Let n be given.
Assume H1.
Apply H1 (λn ⇒ 0 ordsucc n) to the current goal.
We will prove 0 ordsucc 0.
An exact proof term for the current goal is In_0_1.
Let n be given.
Assume IH: 0 ordsucc n.
We will prove 0 ordsucc (ordsucc n).
An exact proof term for the current goal is (ordsuccI1 (ordsucc n) 0 IH).
Theorem. (nat_ordsucc_in_ordsucc)
∀n, nat_p n∀mn, ordsucc m ordsucc n
Proof:
Let n be given.
Assume H1.
Apply (H1 (λn ⇒ ∀mn, ordsucc m ordsucc n)) to the current goal.
We will prove ∀m0, ordsucc m ordsucc 0.
Let m be given.
Assume Hm: m 0.
We will prove False.
An exact proof term for the current goal is (EmptyE m Hm).
Let n be given.
Assume IH: ∀mn, ordsucc m ordsucc n.
We will prove ∀mordsucc n, ordsucc m ordsucc (ordsucc n).
Let m be given.
Assume H2: m ordsucc n.
We will prove ordsucc m ordsucc (ordsucc n).
Apply (ordsuccE n m H2) to the current goal.
Assume H3: m n.
We prove the intermediate claim L1: ordsucc m ordsucc n.
An exact proof term for the current goal is (IH m H3).
An exact proof term for the current goal is (ordsuccI1 (ordsucc n) (ordsucc m) L1).
Assume H3: m = n.
rewrite the current goal using H3 (from left to right).
We will prove ordsucc n ordsucc (ordsucc n).
An exact proof term for the current goal is (ordsuccI2 (ordsucc n)).
Theorem. (nat_ind)
∀p : setprop, p 0(∀n, nat_p np np (ordsucc n))∀n, nat_p np n
Proof:
Let p be given.
Assume H1: p 0.
Assume H2: ∀n, nat_p np np (ordsucc n).
We prove the intermediate claim L1: nat_p 0 p 0.
An exact proof term for the current goal is (andI (nat_p 0) (p 0) nat_0 H1).
We prove the intermediate claim L2: ∀n, nat_p n p nnat_p (ordsucc n) p (ordsucc n).
Let n be given.
Assume H3: nat_p n p n.
Apply H3 to the current goal.
Assume H4: nat_p n.
Assume H5: p n.
Apply andI to the current goal.
We will prove nat_p (ordsucc n).
An exact proof term for the current goal is (nat_ordsucc n H4).
We will prove p (ordsucc n).
An exact proof term for the current goal is (H2 n H4 H5).
Let n be given.
Assume H3.
We prove the intermediate claim L3: nat_p n p n.
An exact proof term for the current goal is (H3 (λn ⇒ nat_p n p n) L1 L2).
An exact proof term for the current goal is (andER (nat_p n) (p n) L3).
Theorem. (nat_complete_ind)
∀p : setprop, (∀n, nat_p n(∀mn, p m)p n)∀n, nat_p np n
Proof:
Let p be given.
Assume H1: ∀n, nat_p n(∀mn, p m)p n.
We prove the intermediate claim L1: ∀n : set, nat_p n∀mn, p m.
Apply nat_ind to the current goal.
We will prove ∀m0, p m.
Let m be given.
Assume Hm: m 0.
We will prove False.
An exact proof term for the current goal is (EmptyE m Hm).
Let n be given.
Assume Hn: nat_p n.
Assume IHn: ∀mn, p m.
We will prove ∀mordsucc n, p m.
Let m be given.
Assume Hm: m ordsucc n.
We will prove p m.
Apply (ordsuccE n m Hm) to the current goal.
Assume H2: m n.
An exact proof term for the current goal is (IHn m H2).
Assume H2: m = n.
We will prove p m.
rewrite the current goal using H2 (from left to right).
We will prove p n.
An exact proof term for the current goal is (H1 n Hn IHn).
We will prove ∀n, nat_p np n.
An exact proof term for the current goal is (λn Hn ⇒ H1 n Hn (L1 n Hn)).
Theorem. (nat_inv_impred)
∀p : setprop, p 0(∀n, nat_p np (ordsucc n))∀n, nat_p np n
Proof:
Let p be given.
Assume H1 H2.
An exact proof term for the current goal is nat_ind p H1 (λn H _ ⇒ H2 n H).
Theorem. (nat_inv)
∀n, nat_p nn = 0 ∃x, nat_p x n = ordsucc x
Proof:
Apply nat_inv_impred to the current goal.
Apply orIL to the current goal.
Use reflexivity.
Let n be given.
Assume Hn.
Apply orIR to the current goal.
We use n to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hn.
Use reflexivity.
Theorem. (nat_p_trans)
∀n, nat_p n∀mn, nat_p m
Proof:
Apply nat_ind to the current goal.
We will prove ∀m0, nat_p m.
Let m be given.
Assume Hm: m 0.
We will prove False.
An exact proof term for the current goal is (EmptyE m Hm).
Let n be given.
Assume Hn: nat_p n.
Assume IHn: ∀mn, nat_p m.
We will prove ∀mordsucc n, nat_p m.
Let m be given.
Assume Hm: m ordsucc n.
Apply (ordsuccE n m Hm) to the current goal.
Assume H1: m n.
An exact proof term for the current goal is (IHn m H1).
Assume H1: m = n.
rewrite the current goal using H1 (from left to right).
An exact proof term for the current goal is Hn.
Theorem. (nat_trans)
∀n, nat_p n∀mn, m n
Proof:
Apply nat_ind to the current goal.
We will prove ∀m0, m 0.
Let m be given.
Assume Hm: m 0.
We will prove False.
An exact proof term for the current goal is (EmptyE m Hm).
Let n be given.
Assume Hn: nat_p n.
Assume IHn: ∀mn, m n.
We will prove ∀mordsucc n, m ordsucc n.
Let m be given.
Assume Hm: m ordsucc n.
Apply (ordsuccE n m Hm) to the current goal.
Assume H1: m n.
We will prove m ordsucc n.
Apply (Subq_tra m n (ordsucc n)) to the current goal.
An exact proof term for the current goal is (IHn m H1).
An exact proof term for the current goal is (ordsuccI1 n).
Assume H1: m = n.
We will prove m ordsucc n.
rewrite the current goal using H1 (from left to right).
We will prove n ordsucc n.
An exact proof term for the current goal is (ordsuccI1 n).
Theorem. (nat_ordsucc_trans)
∀n, nat_p n∀mordsucc n, m n
Proof:
Let n be given.
Assume Hn: nat_p n.
Let m be given.
Assume Hm: m ordsucc n.
Let k be given.
Assume Hk: k m.
We will prove k n.
Apply (ordsuccE n m Hm) to the current goal.
Assume H1: m n.
An exact proof term for the current goal is nat_trans n Hn m H1 k Hk.
Assume H1: m = n.
rewrite the current goal using H1 (from right to left).
An exact proof term for the current goal is Hk.
Definition. We define surj to be λX Y f ⇒ (∀uX, f u Y) (∀wY, ∃uX, f u = w) of type setset(setset)prop.
Theorem. (form100_63_surjCantor)
∀A : set, ∀f : setset, ¬ surj A (𝒫 A) f
Proof:
Let A and f be given.
Assume H.
Apply H to the current goal.
Assume H1: ∀uA, f u 𝒫 A.
Assume H2: ∀w𝒫 A, ∃uA, f u = w.
Set D to be the term {xA|x f x}.
We prove the intermediate claim L1: D 𝒫 A.
An exact proof term for the current goal is Sep_In_Power A (λx ⇒ x f x).
Apply H2 D L1 to the current goal.
Let d be given.
Assume H.
Apply H to the current goal.
Assume Hd: d A.
Assume HfdD: f d = D.
We prove the intermediate claim L2: d D.
Assume H3: d D.
Apply SepE2 A (λx ⇒ x f x) d H3 to the current goal.
We will prove d f d.
rewrite the current goal using HfdD (from left to right).
We will prove d D.
An exact proof term for the current goal is H3.
Apply L2 to the current goal.
We will prove d D.
Apply SepI to the current goal.
We will prove d A.
An exact proof term for the current goal is Hd.
We will prove d f d.
rewrite the current goal using HfdD (from left to right).
An exact proof term for the current goal is L2.
Definition. We define inj to be λX Y f ⇒ (∀uX, f u Y) (∀u vX, f u = f vu = v) of type setset(setset)prop.
Theorem. (form100_63_injCantor)
∀A : set, ∀f : setset, ¬ inj (𝒫 A) A f
Proof:
Let A and f be given.
Assume H.
Apply H to the current goal.
Assume H1: ∀X𝒫 A, f X A.
Assume H2: ∀X Y𝒫 A, f X = f YX = Y.
Set D to be the term {f X|X𝒫 A, f X X}.
We prove the intermediate claim L1: D 𝒫 A.
Apply PowerI to the current goal.
Let n be given.
Assume H3: n D.
Apply ReplSepE_impred (𝒫 A) (λX ⇒ f X X) f n H3 to the current goal.
Let X be given.
Assume HX: X 𝒫 A.
Assume H4: f X X.
Assume H5: n = f X.
We will prove n A.
rewrite the current goal using H5 (from left to right).
Apply H1 to the current goal.
An exact proof term for the current goal is HX.
We prove the intermediate claim L2: f D D.
Assume H3: f D D.
Apply ReplSepE_impred (𝒫 A) (λX ⇒ f X X) f (f D) H3 to the current goal.
Let X be given.
Assume HX: X 𝒫 A.
Assume H4: f X X.
Assume H5: f D = f X.
We prove the intermediate claim L2a: D = X.
An exact proof term for the current goal is H2 D L1 X HX H5.
Apply H4 to the current goal.
rewrite the current goal using L2a (from right to left).
An exact proof term for the current goal is H3.
Apply L2 to the current goal.
We will prove f D D.
Apply ReplSepI to the current goal.
We will prove D 𝒫 A.
An exact proof term for the current goal is L1.
We will prove f D D.
An exact proof term for the current goal is L2.
Theorem. (injI)
∀X Y, ∀f : setset, (∀xX, f x Y)(∀x zX, f x = f zx = z)inj X Y f
Proof:
Let X, Y and f be given.
Assume H1 H2.
We will prove (∀xX, f x Y) (∀x zX, f x = f zx = z).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
Theorem. (inj_comp)
∀X Y Z : set, ∀f g : setset, inj X Y finj Y Z ginj X Z (λx ⇒ g (f x))
Proof:
Let X, Y, Z, f and g be given.
Assume Hf.
Assume Hg.
Apply Hf to the current goal.
Assume Hf1 Hf2.
Apply Hg to the current goal.
Assume Hg1 Hg2.
Apply injI to the current goal.
Let u be given.
Assume Hu: u X.
An exact proof term for the current goal is (Hg1 (f u) (Hf1 u Hu)).
Let u be given.
Assume Hu: u X.
Let v be given.
Assume Hv: v X.
Assume H1: g (f u) = g (f v).
We will prove u = v.
Apply Hf2 u Hu v Hv to the current goal.
We will prove f u = f v.
Apply Hg2 (f u) (Hf1 u Hu) (f v) (Hf1 v Hv) to the current goal.
We will prove g (f u) = g (f v).
An exact proof term for the current goal is H1.
Definition. We define bij to be λX Y f ⇒ (∀uX, f u Y) (∀u vX, f u = f vu = v) (∀wY, ∃uX, f u = w) of type setset(setset)prop.
Theorem. (bijI)
∀X Y, ∀f : setset, (∀uX, f u Y)(∀u vX, f u = f vu = v)(∀wY, ∃uX, f u = w)bij X Y f
Proof:
Let X, Y and f be given.
Assume Hf1 Hf2 Hf3.
We will prove (∀uX, f u Y) (∀u vX, f u = f vu = v) (∀wY, ∃uX, f u = w).
Apply and3I to the current goal.
An exact proof term for the current goal is Hf1.
An exact proof term for the current goal is Hf2.
An exact proof term for the current goal is Hf3.
Theorem. (bijE)
∀X Y, ∀f : setset, bij X Y f∀p : prop, ((∀uX, f u Y)(∀u vX, f u = f vu = v)(∀wY, ∃uX, f u = w)p)p
Proof:
Let X, Y and f be given.
Assume Hf.
Let p be given.
Assume Hp.
Apply Hf to the current goal.
Assume Hf.
Apply Hf to the current goal.
Assume Hf1 Hf2 Hf3.
An exact proof term for the current goal is Hp Hf1 Hf2 Hf3.
Theorem. (bij_inj)
∀X Y, ∀f : setset, bij X Y finj X Y f
Proof:
Let X, Y and f be given.
Assume H1.
Apply H1 to the current goal.
Assume H1 _.
An exact proof term for the current goal is H1.
Theorem. (bij_id)
∀X, bij X X (λx ⇒ x)
Proof:
Let X be given.
We will prove (∀uX, u X) (∀u vX, u = vu = v) (∀wX, ∃uX, u = w).
Apply and3I to the current goal.
An exact proof term for the current goal is (λu Hu ⇒ Hu).
An exact proof term for the current goal is (λu Hu v Hv H1 ⇒ H1).
Let w be given.
Assume Hw.
We use w to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hw.
Use reflexivity.
Theorem. (bij_comp)
∀X Y Z : set, ∀f g : setset, bij X Y fbij Y Z gbij X Z (λx ⇒ g (f x))
Proof:
Let X, Y, Z, f and g be given.
Assume Hf.
Apply Hf to the current goal.
Assume Hf12 Hf3.
Apply Hf12 to the current goal.
Assume Hf1 Hf2.
Assume Hg.
Apply Hg to the current goal.
Assume Hg12 Hg3.
Apply Hg12 to the current goal.
Assume Hg1 Hg2.
We will prove (∀uX, g (f u) Z) (∀u vX, g (f u) = g (f v)u = v) (∀wZ, ∃uX, g (f u) = w).
Apply and3I to the current goal.
Let u be given.
Assume Hu: u X.
An exact proof term for the current goal is (Hg1 (f u) (Hf1 u Hu)).
Let u be given.
Assume Hu: u X.
Let v be given.
Assume Hv: v X.
Assume H1: g (f u) = g (f v).
We will prove u = v.
Apply Hf2 u Hu v Hv to the current goal.
We will prove f u = f v.
Apply Hg2 (f u) (Hf1 u Hu) (f v) (Hf1 v Hv) to the current goal.
We will prove g (f u) = g (f v).
An exact proof term for the current goal is H1.
Let w be given.
Assume Hw: w Z.
Apply Hg3 w Hw to the current goal.
Let y be given.
Assume Hy12.
Apply Hy12 to the current goal.
Assume Hy1: y Y.
Assume Hy2: g y = w.
Apply Hf3 y Hy1 to the current goal.
Let u be given.
Assume Hu12.
Apply Hu12 to the current goal.
Assume Hu1: u X.
Assume Hu2: f u = y.
We will prove ∃uX, g (f u) = w.
We use u to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hu1.
rewrite the current goal using Hu2 (from left to right).
An exact proof term for the current goal is Hy2.
Theorem. (bij_surj)
∀X Y, ∀f : setset, bij X Y fsurj X Y f
Proof:
Let X, Y and f be given.
Assume H1.
Apply H1 to the current goal.
Assume H1.
Apply H1 to the current goal.
Assume H1 _ H2.
We will prove (∀uX, f u Y) (∀wY, ∃uX, f u = w).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
Definition. We define inv to be λX f ⇒ λy : setEps_i (λx ⇒ x X f x = y) of type set(setset)setset.
Theorem. (surj_rinv)
∀X Y, ∀f : setset, (∀wY, ∃uX, f u = w)∀yY, inv X f y X f (inv X f y) = y
Proof:
Let X, Y and f be given.
Assume H1.
Let y be given.
Assume Hy: y Y.
Apply H1 y Hy to the current goal.
Let x be given.
Assume H2.
An exact proof term for the current goal is Eps_i_ax (λx ⇒ x X f x = y) x H2.
Theorem. (inj_linv)
∀X, ∀f : setset, (∀u vX, f u = f vu = v)∀xX, inv X f (f x) = x
Proof:
Let X and f be given.
Assume H1.
Let x be given.
Assume Hx.
We prove the intermediate claim L1: inv X f (f x) X f (inv X f (f x)) = f x.
Apply Eps_i_ax (λx' ⇒ x' X f x' = f x) x to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is Hx.
Use reflexivity.
Apply L1 to the current goal.
Assume H2 H3.
An exact proof term for the current goal is H1 (inv X f (f x)) H2 x Hx H3.
Theorem. (bij_inv)
∀X Y, ∀f : setset, bij X Y fbij Y X (inv X f)
Proof:
Let X, Y and f be given.
Assume H1.
Apply H1 to the current goal.
Assume H2.
Apply H2 to the current goal.
Assume H3: ∀uX, f u Y.
Assume H4: ∀u vX, f u = f vu = v.
Assume H5: ∀wY, ∃uX, f u = w.
Set g to be the term λy ⇒ Eps_i (λx ⇒ x X f x = y) of type setset.
We prove the intermediate claim L1: ∀yY, g y X f (g y) = y.
An exact proof term for the current goal is surj_rinv X Y f H5.
We will prove (∀uY, g u X) (∀u vY, g u = g vu = v) (∀wX, ∃uY, g u = w).
Apply and3I to the current goal.
We will prove ∀uY, g u X.
Let u be given.
Assume Hu.
Apply L1 u Hu to the current goal.
Assume H _.
An exact proof term for the current goal is H.
We will prove ∀u vY, g u = g vu = v.
Let u be given.
Assume Hu.
Let v be given.
Assume Hv H6.
We will prove u = v.
Apply L1 u Hu to the current goal.
Assume H7: g u X.
Assume H8: f (g u) = u.
Apply L1 v Hv to the current goal.
Assume H9: g v X.
Assume H10: f (g v) = v.
rewrite the current goal using H8 (from right to left).
rewrite the current goal using H10 (from right to left).
rewrite the current goal using H6 (from right to left).
Use reflexivity.
We will prove ∀wX, ∃uY, g u = w.
Let w be given.
Assume Hw.
We prove the intermediate claim Lfw: f w Y.
An exact proof term for the current goal is H3 w Hw.
We use f w to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Lfw.
An exact proof term for the current goal is inj_linv X f H4 w Hw.
Definition. We define atleastp to be λX Y : set∃f : setset, inj X Y f of type setsetprop.
Theorem. (atleastp_tra)
∀X Y Z, atleastp X Yatleastp Y Zatleastp X Z
Proof:
The rest of this subproof is missing.
Theorem. (Subq_atleastp)
∀X Y, X Yatleastp X Y
Proof:
The rest of this subproof is missing.
Definition. We define equip to be λX Y : set∃f : setset, bij X Y f of type setsetprop.
Theorem. (equip_atleastp)
∀X Y, equip X Yatleastp X Y
Proof:
The rest of this subproof is missing.
Theorem. (equip_ref)
∀X, equip X X
Proof:
The rest of this subproof is missing.
Theorem. (equip_sym)
∀X Y, equip X Yequip Y X
Proof:
The rest of this subproof is missing.
Theorem. (equip_tra)
∀X Y Z, equip X Yequip Y Zequip X Z
Proof:
The rest of this subproof is missing.
Theorem. (equip_0_Empty)
∀X, equip X 0X = 0
Proof:
The rest of this subproof is missing.
Theorem. (equip_adjoin_ordsucc)
∀N X y, y Xequip N Xequip (ordsucc N) (X {y})
Proof:
The rest of this subproof is missing.
Theorem. (equip_ordsucc_remove1)
∀X N, ∀xX, equip X (ordsucc N)equip (X {x}) N
Proof:
The rest of this subproof is missing.
Beginning of Section SchroederBernstein
Theorem. (KnasterTarski_set)
∀A, ∀F : setset, (∀U𝒫 A, F U 𝒫 A)(∀U V𝒫 A, U VF U F V)∃Y𝒫 A, F Y = Y
Proof:
The rest of this subproof is missing.
Theorem. (image_In_Power)
∀A B, ∀f : setset, (∀xA, f x B)∀U𝒫 A, {f x|xU} 𝒫 B
Proof:
The rest of this subproof is missing.
Theorem. (image_monotone)
∀f : setset, ∀U V, U V{f x|xU} {f x|xV}
Proof:
The rest of this subproof is missing.
Theorem. (setminus_antimonotone)
∀A U V, U VA V A U
Proof:
The rest of this subproof is missing.
Theorem. (SchroederBernstein)
∀A B, ∀f g : setset, inj A B finj B A gequip A B
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section SchroederBernstein
Beginning of Section PigeonHole
Theorem. (PigeonHole_nat)
∀n, nat_p n∀f : setset, (∀iordsucc n, f i n)¬ (∀i jordsucc n, f i = f ji = j)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section PigeonHole
Theorem. (Union_ordsucc_eq)
∀n, nat_p n (ordsucc n) = n
Proof:
The rest of this subproof is missing.
Theorem. (cases_1)
∀i1, ∀p : setprop, p 0p i
Proof:
The rest of this subproof is missing.
Theorem. (cases_2)
∀i2, ∀p : setprop, p 0p 1p i
Proof:
The rest of this subproof is missing.
Theorem. (neq_0_1)
Proof:
The rest of this subproof is missing.
Theorem. (neq_1_0)
Proof:
The rest of this subproof is missing.
Theorem. (neq_0_2)
Proof:
The rest of this subproof is missing.
Theorem. (neq_2_0)
Proof:
The rest of this subproof is missing.
Definition. We define ordinal to be λalpha : setTransSet alpha ∀betaalpha, TransSet beta of type setprop.
Theorem. (ordinal_TransSet)
∀alpha : set, ordinal alphaTransSet alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_Empty)
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_Hered)
∀alpha : set, ordinal alpha∀betaalpha, ordinal beta
Proof:
The rest of this subproof is missing.
Theorem. (TransSet_ordsucc)
∀X : set, TransSet XTransSet (ordsucc X)
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_ordsucc)
∀alpha : set, ordinal alphaordinal (ordsucc alpha)
Proof:
The rest of this subproof is missing.
Theorem. (nat_p_ordinal)
∀n : set, nat_p nordinal n
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_1)
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_2)
Proof:
The rest of this subproof is missing.
Theorem. (TransSet_ordsucc_In_Subq)
∀X : set, TransSet X∀xX, ordsucc x X
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_ordsucc_In_Subq)
∀alpha, ordinal alpha∀betaalpha, ordsucc beta alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_trichotomy_or)
∀alpha beta : set, ordinal alphaordinal betaalpha beta alpha = beta beta alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_trichotomy_or_impred)
∀alpha beta : set, ordinal alphaordinal beta∀p : prop, (alpha betap)(alpha = betap)(beta alphap)p
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_In_Or_Subq)
∀alpha beta, ordinal alphaordinal betaalpha beta beta alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_linear)
∀alpha beta, ordinal alphaordinal betaalpha beta beta alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_ordsucc_In_eq)
∀alpha beta, ordinal alphabeta alphaordsucc beta alpha alpha = ordsucc beta
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_lim_or_succ)
∀alpha, ordinal alpha(∀betaalpha, ordsucc beta alpha) (∃betaalpha, alpha = ordsucc beta)
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_ordsucc_In)
∀alpha, ordinal alpha∀betaalpha, ordsucc beta ordsucc alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_famunion)
∀X, ∀F : setset, (∀xX, ordinal (F x))ordinal (xXF x)
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_binintersect)
∀alpha beta, ordinal alphaordinal betaordinal (alpha beta)
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_binunion)
∀alpha beta, ordinal alphaordinal betaordinal (alpha beta)
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_ind)
∀p : setprop, (∀alpha, ordinal alpha(∀betaalpha, p beta)p alpha)∀alpha, ordinal alphap alpha
Proof:
The rest of this subproof is missing.
Theorem. (least_ordinal_ex)
∀p : setprop, (∃alpha, ordinal alpha p alpha)∃alpha, ordinal alpha p alpha ∀betaalpha, ¬ p beta
Proof:
The rest of this subproof is missing.
Theorem. (equip_Sing_1)
∀x, equip {x} 1
Proof:
The rest of this subproof is missing.
Theorem. (TransSet_In_ordsucc_Subq)
∀x y, TransSet yx ordsucc yx y
Proof:
The rest of this subproof is missing.
Theorem. (exandE_i)
∀P Q : setprop, (∃x, P x Q x)∀r : prop, (∀x, P xQ xr)r
Proof:
The rest of this subproof is missing.
Theorem. (exandE_ii)
∀P Q : (setset)prop, (∃x : setset, P x Q x)∀p : prop, (∀x : setset, P xQ xp)p
Proof:
The rest of this subproof is missing.
Theorem. (exandE_iii)
∀P Q : (setsetset)prop, (∃x : setsetset, P x Q x)∀p : prop, (∀x : setsetset, P xQ xp)p
Proof:
The rest of this subproof is missing.
Theorem. (exandE_iiii)
∀P Q : (setsetsetset)prop, (∃x : setsetsetset, P x Q x)∀p : prop, (∀x : setsetsetset, P xQ xp)p
Proof:
The rest of this subproof is missing.
Beginning of Section Descr_ii
Variable P : (setset)prop
Definition. We define Descr_ii to be λx : setEps_i (λy ⇒ ∀h : setset, P hh x = y) of type setset.
Hypothesis Pex : ∃f : setset, P f
Hypothesis Puniq : ∀f g : setset, P fP gf = g
Proof:
The rest of this subproof is missing.
End of Section Descr_ii
Beginning of Section Descr_iii
Variable P : (setsetset)prop
Definition. We define Descr_iii to be λx y : setEps_i (λz ⇒ ∀h : setsetset, P hh x y = z) of type setsetset.
Hypothesis Pex : ∃f : setsetset, P f
Hypothesis Puniq : ∀f g : setsetset, P fP gf = g
Proof:
The rest of this subproof is missing.
End of Section Descr_iii
Beginning of Section Descr_Vo1
Variable P : Vo 1prop
Definition. We define Descr_Vo1 to be λx : set∀h : setprop, P hh x of type Vo 1.
Hypothesis Pex : ∃f : Vo 1, P f
Hypothesis Puniq : ∀f g : Vo 1, P fP gf = g
Proof:
The rest of this subproof is missing.
End of Section Descr_Vo1
Beginning of Section If_ii
Variable p : prop
Variable f g : setset
Definition. We define If_ii to be λx ⇒ if p then f x else g x of type setset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section If_ii
Beginning of Section If_iii
Variable p : prop
Variable f g : setsetset
Definition. We define If_iii to be λx y ⇒ if p then f x y else g x y of type setsetset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section If_iii
Beginning of Section EpsilonRec_i
Variable F : set(setset)set
Definition. We define In_rec_i_G to be λX Y ⇒ ∀R : setsetprop, (∀X : set, ∀f : setset, (∀xX, R x (f x))R X (F X f))R X Y of type setsetprop.
Definition. We define In_rec_i to be λX ⇒ Eps_i (In_rec_i_G X) of type setset.
Theorem. (In_rec_i_G_c)
∀X : set, ∀f : setset, (∀xX, In_rec_i_G x (f x))In_rec_i_G X (F X f)
Proof:
The rest of this subproof is missing.
Theorem. (In_rec_i_G_inv)
∀X : set, ∀Y : set, In_rec_i_G X Y∃f : setset, (∀xX, In_rec_i_G x (f x)) Y = F X f
Proof:
The rest of this subproof is missing.
Hypothesis Fr : ∀X : set, ∀g h : setset, (∀xX, g x = h x)F X g = F X h
Theorem. (In_rec_i_G_f)
∀X : set, ∀Y Z : set, In_rec_i_G X YIn_rec_i_G X ZY = Z
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section EpsilonRec_i
Beginning of Section EpsilonRec_ii
Variable F : set(set(setset))(setset)
Definition. We define In_rec_G_ii to be λX Y ⇒ ∀R : set(setset)prop, (∀X : set, ∀f : set(setset), (∀xX, R x (f x))R X (F X f))R X Y of type set(setset)prop.
Definition. We define In_rec_ii to be λX ⇒ Descr_ii (In_rec_G_ii X) of type set(setset).
Theorem. (In_rec_G_ii_c)
∀X : set, ∀f : set(setset), (∀xX, In_rec_G_ii x (f x))In_rec_G_ii X (F X f)
Proof:
The rest of this subproof is missing.
Theorem. (In_rec_G_ii_inv)
∀X : set, ∀Y : (setset), In_rec_G_ii X Y∃f : set(setset), (∀xX, In_rec_G_ii x (f x)) Y = F X f
Proof:
The rest of this subproof is missing.
Hypothesis Fr : ∀X : set, ∀g h : set(setset), (∀xX, g x = h x)F X g = F X h
Theorem. (In_rec_G_ii_f)
∀X : set, ∀Y Z : (setset), In_rec_G_ii X YIn_rec_G_ii X ZY = Z
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section EpsilonRec_ii
Beginning of Section EpsilonRec_iii
Variable F : set(set(setsetset))(setsetset)
Definition. We define In_rec_G_iii to be λX Y ⇒ ∀R : set(setsetset)prop, (∀X : set, ∀f : set(setsetset), (∀xX, R x (f x))R X (F X f))R X Y of type set(setsetset)prop.
Definition. We define In_rec_iii to be λX ⇒ Descr_iii (In_rec_G_iii X) of type set(setsetset).
Theorem. (In_rec_G_iii_c)
∀X : set, ∀f : set(setsetset), (∀xX, In_rec_G_iii x (f x))In_rec_G_iii X (F X f)
Proof:
The rest of this subproof is missing.
Theorem. (In_rec_G_iii_inv)
∀X : set, ∀Y : (setsetset), In_rec_G_iii X Y∃f : set(setsetset), (∀xX, In_rec_G_iii x (f x)) Y = F X f
Proof:
The rest of this subproof is missing.
Hypothesis Fr : ∀X : set, ∀g h : set(setsetset), (∀xX, g x = h x)F X g = F X h
Theorem. (In_rec_G_iii_f)
∀X : set, ∀Y Z : (setsetset), In_rec_G_iii X YIn_rec_G_iii X ZY = Z
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section EpsilonRec_iii
Beginning of Section NatRec
Variable z : set
Variable f : setsetset
Let F : set(setset)setλn g ⇒ if n n then f ( n) (g ( n)) else z
Definition. We define nat_primrec to be In_rec_i F of type setset.
Theorem. (nat_primrec_r)
∀X : set, ∀g h : setset, (∀xX, g x = h x)F X g = F X h
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section NatRec
Beginning of Section NatAdd
Definition. We define add_nat to be λn m : setnat_primrec n (λ_ r ⇒ ordsucc r) m of type setsetset.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Theorem. (add_nat_0R)
∀n : set, n + 0 = n
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_SR)
∀n m : set, nat_p mn + ordsucc m = ordsucc (n + m)
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_p)
∀n : set, nat_p n∀m : set, nat_p mnat_p (n + m)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_asso)
∀n : set, nat_p n∀m : set, nat_p m∀k : set, nat_p k(n + m) + k = n + (m + k)
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_0L)
∀m : set, nat_p m0 + m = m
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_SL)
∀n : set, nat_p n∀m : set, nat_p mordsucc n + m = ordsucc (n + m)
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_com)
∀n : set, nat_p n∀m : set, nat_p mn + m = m + n
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_In_R)
∀m, nat_p m∀km, ∀n, nat_p nk + n m + n
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_In_L)
∀n, nat_p n∀m, nat_p m∀km, n + k n + m
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_Subq_R)
∀k, nat_p k∀m, nat_p mk m∀n, nat_p nk + n m + n
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_Subq_L)
∀n, nat_p n∀k, nat_p k∀m, nat_p mk mn + k n + m
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_Subq_R')
∀m, nat_p m∀n, nat_p nm m + n
Proof:
The rest of this subproof is missing.
Theorem. (nat_Subq_add_ex)
∀n, nat_p n∀m, nat_p mn m∃k, nat_p k m = k + n
Proof:
The rest of this subproof is missing.
Theorem. (add_nat_cancel_R)
∀k, nat_p k∀m, nat_p m∀n, nat_p nk + n = m + nk = m
Proof:
The rest of this subproof is missing.
End of Section NatAdd
Beginning of Section NatMul
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Definition. We define mul_nat to be λn m : setnat_primrec 0 (λ_ r ⇒ n + r) m of type setsetset.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Theorem. (mul_nat_0R)
∀n : set, n * 0 = 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_SR)
∀n m, nat_p mn * ordsucc m = n + n * m
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_p)
∀n : set, nat_p n∀m : set, nat_p mnat_p (n * m)
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_0L)
∀m : set, nat_p m0 * m = 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_SL)
∀n : set, nat_p n∀m : set, nat_p mordsucc n * m = n * m + m
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_com)
∀n : set, nat_p n∀m : set, nat_p mn * m = m * n
Proof:
The rest of this subproof is missing.
Theorem. (mul_add_nat_distrL)
∀n : set, nat_p n∀m : set, nat_p m∀k : set, nat_p kn * (m + k) = n * m + n * k
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_asso)
∀n : set, nat_p n∀m : set, nat_p m∀k : set, nat_p k(n * m) * k = n * (m * k)
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_Subq_R)
∀m n, nat_p mnat_p nm n∀k, nat_p km * k n * k
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_Subq_L)
∀m n, nat_p mnat_p nm n∀k, nat_p kk * m k * n
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_0_or_Subq)
∀m, nat_p m∀n, nat_p nn = 0 m m * n
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_0_inv)
∀m, nat_p m∀n, nat_p nm * n = 0m = 0 n = 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_nat_0m_1n_In)
∀m, nat_p m∀n, nat_p n0 m1 nm m * n
Proof:
The rest of this subproof is missing.
Theorem. (nat_le1_cases)
∀m, nat_p mm 1m = 0 m = 1
Proof:
The rest of this subproof is missing.
Definition. We define Pi_nat to be λf n ⇒ nat_primrec 1 (λi r ⇒ r * f i) n of type (setset)setset.
Theorem. (Pi_nat_0)
∀f : setset, Pi_nat f 0 = 1
Proof:
The rest of this subproof is missing.
Theorem. (Pi_nat_S)
∀f : setset, ∀n, nat_p nPi_nat f (ordsucc n) = Pi_nat f n * f n
Proof:
The rest of this subproof is missing.
Theorem. (Pi_nat_p)
∀f : setset, ∀n, nat_p n(∀in, nat_p (f i))nat_p (Pi_nat f n)
Proof:
The rest of this subproof is missing.
Theorem. (Pi_nat_0_inv)
∀f : setset, ∀n, nat_p n(∀in, nat_p (f i))Pi_nat f n = 0(∃in, f i = 0)
Proof:
The rest of this subproof is missing.
Definition. We define exp_nat to be λn m : setnat_primrec 1 (λ_ r ⇒ n * r) m of type setsetset.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_nat.
Proof:
The rest of this subproof is missing.
Theorem. (exp_nat_S)
∀n m, nat_p mn ^ (ordsucc m) = n * n ^ m
Proof:
The rest of this subproof is missing.
Theorem. (exp_nat_p)
∀n, nat_p n∀m, nat_p mnat_p (n ^ m)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section NatMul
Beginning of Section form100_52
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_nat.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (equip_finite_Power)
∀n, nat_p n∀X, equip X nequip (𝒫 X) (2 ^ n)
Proof:
The rest of this subproof is missing.
End of Section form100_52
Theorem. (ZF_closed_E)
∀U, ZF_closed U∀p : prop, (Union_closed UPower_closed URepl_closed Up)p
Proof:
The rest of this subproof is missing.
Theorem. (ZF_Union_closed)
∀U, ZF_closed U∀XU, X U
Proof:
The rest of this subproof is missing.
Theorem. (ZF_Power_closed)
∀U, ZF_closed U∀XU, 𝒫 X U
Proof:
The rest of this subproof is missing.
Theorem. (ZF_Repl_closed)
∀U, ZF_closed U∀XU, ∀F : setset, (∀xX, F x U){F x|xX} U
Proof:
The rest of this subproof is missing.
Theorem. (ZF_UPair_closed)
∀U, ZF_closed U∀x yU, {x,y} U
Proof:
The rest of this subproof is missing.
Theorem. (ZF_Sing_closed)
∀U, ZF_closed U∀xU, {x} U
Proof:
The rest of this subproof is missing.
Theorem. (ZF_binunion_closed)
∀U, ZF_closed U∀X YU, (X Y) U
Proof:
The rest of this subproof is missing.
Theorem. (ZF_ordsucc_closed)
∀U, ZF_closed U∀xU, ordsucc x U
Proof:
The rest of this subproof is missing.
Theorem. (nat_p_UnivOf_Empty)
∀n : set, nat_p nn UnivOf Empty
Proof:
The rest of this subproof is missing.
Definition. We define ω to be {nUnivOf Empty|nat_p n} of type set.
Theorem. (omega_nat_p)
∀nω, nat_p n
Proof:
The rest of this subproof is missing.
Theorem. (nat_p_omega)
∀n : set, nat_p nn ω
Proof:
The rest of this subproof is missing.
Theorem. (omega_ordsucc)
Proof:
The rest of this subproof is missing.
Theorem. (form100_22_v2)
∀f : setset, ¬ inj (𝒫 ω) ω f
Proof:
The rest of this subproof is missing.
Theorem. (form100_22_v3)
∀g : setset, ¬ surj ω (𝒫 ω) g
Proof:
The rest of this subproof is missing.
Theorem. (form100_22_v1)
Proof:
The rest of this subproof is missing.
Theorem. (omega_TransSet)
Proof:
The rest of this subproof is missing.
Theorem. (omega_ordinal)
Proof:
The rest of this subproof is missing.
Theorem. (ordsucc_omega_ordinal)
Proof:
The rest of this subproof is missing.
Definition. We define finite to be λX ⇒ ∃nω, equip X n of type setprop.
Theorem. (nat_finite)
∀n, nat_p nfinite n
Proof:
The rest of this subproof is missing.
Theorem. (finite_ind)
∀p : setprop, p Empty(∀X y, finite Xy Xp Xp (X {y}))∀X, finite Xp X
Proof:
The rest of this subproof is missing.
Theorem. (finite_Empty)
Proof:
The rest of this subproof is missing.
Theorem. (Sing_finite)
∀x, finite {x}
Proof:
The rest of this subproof is missing.
Theorem. (adjoin_finite)
∀X y, finite Xfinite (X {y})
Proof:
The rest of this subproof is missing.
Theorem. (binunion_finite)
∀X, finite X∀Y, finite Yfinite (X Y)
Proof:
The rest of this subproof is missing.
Theorem. (famunion_nat_finite)
∀X : setset, ∀n, nat_p n(∀in, finite (X i))finite (inX i)
Proof:
The rest of this subproof is missing.
Theorem. (Subq_finite)
∀X, finite X∀Y, Y Xfinite Y
Proof:
The rest of this subproof is missing.
Definition. We define infinite to be λA ⇒ ¬ finite A of type setprop.
Beginning of Section InfinitePrimes
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Definition. We define divides_nat to be λm n ⇒ m ω n ω ∃kω, m * k = n of type setsetprop.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define prime_nat to be λn ⇒ n ω 1 n ∀kω, divides_nat k nk = 1 k = n of type setprop.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (Pi_nat_divides)
∀f : setset, ∀n, nat_p n(∀in, nat_p (f i))(∀in, divides_nat (f i) (Pi_nat f n))
Proof:
The rest of this subproof is missing.
Definition. We define composite_nat to be λn ⇒ n ω ∃k mω, 1 k 1 m k * m = n of type setprop.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define primes to be {nω|prime_nat n} of type set.
Proof:
The rest of this subproof is missing.
End of Section InfinitePrimes
Beginning of Section InfiniteRamsey
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (infinite_Finite_Subq_ex)
∀X, infinite X∀n, nat_p n∃YX, equip Y n
Proof:
The rest of this subproof is missing.
Theorem. (infiniteRamsey_lem)
∀X, ∀f g f' : setset, infinite X(∀ZX, infinite Zf Z Z infinite (f Z))(∀ZX, infinite Zg Z Z g Z f Z)f' 0 = f X(∀m, nat_p mf' (ordsucc m) = f (f' m))(∀m, nat_p mf' m X infinite (f' m)) (∀m m'ω, m m'f' m' f' m) (∀m m'ω, g (f' m) = g (f' m')m = m')
Proof:
The rest of this subproof is missing.
Theorem. (infiniteRamsey)
∀c, nat_p c∀n, nat_p n∀X, infinite X∀C : setset, (∀YX, equip Y nC Y c)∃HX, ∃ic, infinite H ∀YH, equip Y nC Y = i
Proof:
The rest of this subproof is missing.
End of Section InfiniteRamsey
Definition. We define Inj1 to be In_rec_i (λX f ⇒ {0} {f x|xX}) of type setset.
Theorem. (Inj1_eq)
∀X : set, Inj1 X = {0} {Inj1 x|xX}
Proof:
The rest of this subproof is missing.
Theorem. (Inj1I1)
∀X : set, 0 Inj1 X
Proof:
The rest of this subproof is missing.
Theorem. (Inj1I2)
∀X x : set, x XInj1 x Inj1 X
Proof:
The rest of this subproof is missing.
Theorem. (Inj1E)
∀X y : set, y Inj1 Xy = 0 ∃xX, y = Inj1 x
Proof:
The rest of this subproof is missing.
Theorem. (Inj1NE1)
∀x : set, Inj1 x 0
Proof:
The rest of this subproof is missing.
Theorem. (Inj1NE2)
∀x : set, Inj1 x {0}
Proof:
The rest of this subproof is missing.
Definition. We define Inj0 to be λX ⇒ {Inj1 x|xX} of type setset.
Theorem. (Inj0I)
∀X x : set, x XInj1 x Inj0 X
Proof:
The rest of this subproof is missing.
Theorem. (Inj0E)
∀X y : set, y Inj0 X∃x : set, x X y = Inj1 x
Proof:
The rest of this subproof is missing.
Definition. We define Unj to be In_rec_i (λX f ⇒ {f x|xX {0}}) of type setset.
Theorem. (Unj_eq)
∀X : set, Unj X = {Unj x|xX {0}}
Proof:
The rest of this subproof is missing.
Theorem. (Unj_Inj1_eq)
∀X : set, Unj (Inj1 X) = X
Proof:
The rest of this subproof is missing.
Theorem. (Inj1_inj)
∀X Y : set, Inj1 X = Inj1 YX = Y
Proof:
The rest of this subproof is missing.
Theorem. (Unj_Inj0_eq)
∀X : set, Unj (Inj0 X) = X
Proof:
The rest of this subproof is missing.
Theorem. (Inj0_inj)
∀X Y : set, Inj0 X = Inj0 YX = Y
Proof:
The rest of this subproof is missing.
Theorem. (Inj0_0)
Proof:
The rest of this subproof is missing.
Theorem. (Inj0_Inj1_neq)
∀X Y : set, Inj0 X Inj1 Y
Proof:
The rest of this subproof is missing.
Definition. We define setsum to be λX Y ⇒ {Inj0 x|xX} {Inj1 y|yY} of type setsetset.
Notation. We use + as an infix operator with priority 450 and which associates to the left corresponding to applying term setsum.
Theorem. (Inj0_setsum)
∀X Y x : set, x XInj0 x X + Y
Proof:
The rest of this subproof is missing.
Theorem. (Inj1_setsum)
∀X Y y : set, y YInj1 y X + Y
Proof:
The rest of this subproof is missing.
Theorem. (setsum_Inj_inv)
∀X Y z : set, z X + Y(∃xX, z = Inj0 x) (∃yY, z = Inj1 y)
Proof:
The rest of this subproof is missing.
Theorem. (Inj0_setsum_0L)
∀X : set, 0 + X = Inj0 X
Proof:
The rest of this subproof is missing.
Theorem. (Inj1_setsum_1L)
∀X : set, 1 + X = Inj1 X
Proof:
The rest of this subproof is missing.
Beginning of Section pair_setsum
Let pair ≝ setsum
Definition. We define proj0 to be λZ ⇒ {Unj z|zZ, ∃x : set, Inj0 x = z} of type setset.
Definition. We define proj1 to be λZ ⇒ {Unj z|zZ, ∃y : set, Inj1 y = z} of type setset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (pairI0)
∀X Y x, x Xpair 0 x pair X Y
Proof:
The rest of this subproof is missing.
Theorem. (pairI1)
∀X Y y, y Ypair 1 y pair X Y
Proof:
The rest of this subproof is missing.
Theorem. (pairE)
∀X Y z, z pair X Y(∃xX, z = pair 0 x) (∃yY, z = pair 1 y)
Proof:
The rest of this subproof is missing.
Theorem. (pairE0)
∀X Y x, pair 0 x pair X Yx X
Proof:
The rest of this subproof is missing.
Theorem. (pairE1)
∀X Y y, pair 1 y pair X Yy Y
Proof:
The rest of this subproof is missing.
Theorem. (proj0I)
∀w u : set, pair 0 u wu proj0 w
Proof:
The rest of this subproof is missing.
Theorem. (proj0E)
∀w u : set, u proj0 wpair 0 u w
Proof:
The rest of this subproof is missing.
Theorem. (proj1I)
∀w u : set, pair 1 u wu proj1 w
Proof:
The rest of this subproof is missing.
Theorem. (proj1E)
∀w u : set, u proj1 wpair 1 u w
Proof:
The rest of this subproof is missing.
Theorem. (proj0_pair_eq)
∀X Y : set, proj0 (pair X Y) = X
Proof:
The rest of this subproof is missing.
Theorem. (proj1_pair_eq)
∀X Y : set, proj1 (pair X Y) = Y
Proof:
The rest of this subproof is missing.
Definition. We define Sigma to be λX Y ⇒ xX{pair x y|yY x} of type set(setset)set.
Notation. We use x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Sigma.
Theorem. (Sigma_eta_proj0_proj1)
∀X : set, ∀Y : setset, ∀z(xX, Y x), pair (proj0 z) (proj1 z) = z proj0 z X proj1 z Y (proj0 z)
Proof:
The rest of this subproof is missing.
Theorem. (proj0_Sigma)
∀X : set, ∀Y : setset, ∀z : set, z (xX, Y x)proj0 z X
Proof:
The rest of this subproof is missing.
Theorem. (proj1_Sigma)
∀X : set, ∀Y : setset, ∀z : set, z (xX, Y x)proj1 z Y (proj0 z)
Proof:
The rest of this subproof is missing.
Theorem. (pair_Sigma)
∀X : set, ∀Y : setset, ∀xX, ∀yY x, pair x y xX, Y x
Proof:
The rest of this subproof is missing.
Theorem. (pair_Sigma_E1)
∀X : set, ∀Y : setset, ∀x y : set, pair x y (xX, Y x)y Y x
Proof:
The rest of this subproof is missing.
Theorem. (Sigma_E)
∀X : set, ∀Y : setset, ∀z : set, z (xX, Y x)∃xX, ∃yY x, z = pair x y
Proof:
The rest of this subproof is missing.
Definition. We define setprod to be λX Y : setxX, Y of type setsetset.
Notation. We use as an infix operator with priority 440 and which associates to the left corresponding to applying term setprod.
Let lam : set(setset)setSigma
Definition. We define ap to be λf x ⇒ {proj1 z|zf, ∃y : set, z = pair x y} of type setsetset.
Notation. When x is a set, a term x y is notation for ap x y.
Notation. λ xAB is notation for the set Sigma Ax : set ⇒ B).
Notation. We now use n-tuple notation (a0,...,an-1) for n ≥ 2 for λ i ∈ n . if i = 0 then a0 else ... if i = n-2 then an-2 else an-1.
Theorem. (lamI)
∀X : set, ∀F : setset, ∀xX, ∀yF x, pair x y λxXF x
Proof:
The rest of this subproof is missing.
Theorem. (lamE)
∀X : set, ∀F : setset, ∀z : set, z (λxXF x)∃xX, ∃yF x, z = pair x y
Proof:
The rest of this subproof is missing.
Theorem. (apI)
∀f x y, pair x y fy f x
Proof:
The rest of this subproof is missing.
Theorem. (apE)
∀f x y, y f xpair x y f
Proof:
The rest of this subproof is missing.
Theorem. (beta)
∀X : set, ∀F : setset, ∀x : set, x X(λxXF x) x = F x
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (pair_ap_0)
∀x y : set, (pair x y) 0 = x
Proof:
The rest of this subproof is missing.
Theorem. (pair_ap_1)
∀x y : set, (pair x y) 1 = y
Proof:
The rest of this subproof is missing.
Theorem. (ap0_Sigma)
∀X : set, ∀Y : setset, ∀z : set, z (xX, Y x)(z 0) X
Proof:
The rest of this subproof is missing.
Theorem. (ap1_Sigma)
∀X : set, ∀Y : setset, ∀z : set, z (xX, Y x)(z 1) (Y (z 0))
Proof:
The rest of this subproof is missing.
Definition. We define pair_p to be λu : setpair (u 0) (u 1) = u of type setprop.
Theorem. (pair_p_I)
∀x y, pair_p (pair x y)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (tuple_pair)
∀x y : set, pair x y = (x,y)
Proof:
The rest of this subproof is missing.
Definition. We define Pi to be λX Y ⇒ {f𝒫 (xX, (Y x))|∀xX, f x Y x} of type set(setset)set.
Notation. We use x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Pi.
Theorem. (PiI)
∀X : set, ∀Y : setset, ∀f : set, (∀uf, pair_p u u 0 X)(∀xX, f x Y x)f xX, Y x
Proof:
The rest of this subproof is missing.
Theorem. (lam_Pi)
∀X : set, ∀Y : setset, ∀F : setset, (∀xX, F x Y x)(λxXF x) (xX, Y x)
Proof:
The rest of this subproof is missing.
Theorem. (ap_Pi)
∀X : set, ∀Y : setset, ∀f : set, ∀x : set, f (xX, Y x)x Xf x Y x
Proof:
The rest of this subproof is missing.
Definition. We define setexp to be λX Y : setyY, X of type setsetset.
Notation. We use :^: as an infix operator with priority 430 and which associates to the left corresponding to applying term setexp.
Theorem. (pair_tuple_fun)
pair = (λx y ⇒ (x,y))
Proof:
The rest of this subproof is missing.
Beginning of Section Tuples
Variable x0 x1 : set
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section Tuples
Theorem. (ReplEq_setprod_ext)
∀X Y, ∀F G : setsetset, (∀xX, ∀yY, F x y = G x y){F (w 0) (w 1)|wX Y} = {G (w 0) (w 1)|wX Y}
Proof:
The rest of this subproof is missing.
Theorem. (lamI2)
∀X, ∀F : setset, ∀xX, ∀yF x, (x,y) λxXF x
Proof:
The rest of this subproof is missing.
Theorem. (tuple_2_Sigma)
∀X : set, ∀Y : setset, ∀xX, ∀yY x, (x,y) xX, Y x
Proof:
The rest of this subproof is missing.
Theorem. (tuple_2_setprod)
∀X : set, ∀Y : set, ∀xX, ∀yY, (x,y) X Y
Proof:
The rest of this subproof is missing.
End of Section pair_setsum
Notation. We use x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Sigma.
Notation. We use as an infix operator with priority 440 and which associates to the left corresponding to applying term setprod.
Notation. We use x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Pi.
Notation. We use :^: as an infix operator with priority 430 and which associates to the left corresponding to applying term setexp.
Definition. We define DescrR_i_io_1 to be λR ⇒ Eps_i (λx ⇒ (∃y : setprop, R x y) (∀y z : setprop, R x yR x zy = z)) of type (set(setprop)prop)set.
Definition. We define DescrR_i_io_2 to be λR ⇒ Descr_Vo1 (λy ⇒ R (DescrR_i_io_1 R) y) of type (set(setprop)prop)setprop.
Theorem. (DescrR_i_io_12)
∀R : set(setprop)prop, (∃x, (∃y : setprop, R x y) (∀y z : setprop, R x yR x zy = z))R (DescrR_i_io_1 R) (DescrR_i_io_2 R)
Proof:
The rest of this subproof is missing.
Definition. We define PNoEq_ to be λalpha p q ⇒ ∀betaalpha, p beta q beta of type set(setprop)(setprop)prop.
Theorem. (PNoEq_ref_)
∀alpha, ∀p : setprop, PNoEq_ alpha p p
Proof:
The rest of this subproof is missing.
Theorem. (PNoEq_sym_)
∀alpha, ∀p q : setprop, PNoEq_ alpha p qPNoEq_ alpha q p
Proof:
The rest of this subproof is missing.
Theorem. (PNoEq_tra_)
∀alpha, ∀p q r : setprop, PNoEq_ alpha p qPNoEq_ alpha q rPNoEq_ alpha p r
Proof:
The rest of this subproof is missing.
Theorem. (PNoEq_antimon_)
∀p q : setprop, ∀alpha, ordinal alpha∀betaalpha, PNoEq_ alpha p qPNoEq_ beta p q
Proof:
The rest of this subproof is missing.
Definition. We define PNoLt_ to be λalpha p q ⇒ ∃betaalpha, PNoEq_ beta p q ¬ p beta q beta of type set(setprop)(setprop)prop.
Theorem. (PNoLt_E_)
∀alpha, ∀p q : setprop, PNoLt_ alpha p q∀R : prop, (∀beta, beta alphaPNoEq_ beta p q¬ p betaq betaR)R
Proof:
The rest of this subproof is missing.
Theorem. (PNoLt_irref_)
∀alpha, ∀p : setprop, ¬ PNoLt_ alpha p p
Proof:
The rest of this subproof is missing.
Theorem. (PNoLt_mon_)
∀p q : setprop, ∀alpha, ordinal alpha∀betaalpha, PNoLt_ beta p qPNoLt_ alpha p q
Proof:
The rest of this subproof is missing.
Theorem. (PNoLt_trichotomy_or_)
∀p q : setprop, ∀alpha, ordinal alphaPNoLt_ alpha p q PNoEq_ alpha p q PNoLt_ alpha q p
Proof:
The rest of this subproof is missing.
Definition. We define PNoLt to be λalpha p beta q ⇒ PNoLt_ (alpha beta) p q alpha beta PNoEq_ alpha p q q alpha beta alpha PNoEq_ beta p q ¬ p beta of type set(setprop)set(setprop)prop.
Theorem. (PNoLtI1)
∀alpha beta, ∀p q : setprop, PNoLt_ (alpha beta) p qPNoLt alpha p beta q
Proof:
The rest of this subproof is missing.
Theorem. (PNoLtI2)
∀alpha beta, ∀p q : setprop, alpha betaPNoEq_ alpha p qq alphaPNoLt alpha p beta q
Proof:
The rest of this subproof is missing.
Theorem. (PNoLtI3)
∀alpha beta, ∀p q : setprop, beta alphaPNoEq_ beta p q¬ p betaPNoLt alpha p beta q
Proof:
The rest of this subproof is missing.
Theorem. (PNoLtE)
∀alpha beta, ∀p q : setprop, PNoLt alpha p beta q∀R : prop, (PNoLt_ (alpha beta) p qR)(alpha betaPNoEq_ alpha p qq alphaR)(beta alphaPNoEq_ beta p q¬ p betaR)R
Proof:
The rest of this subproof is missing.
Theorem. (PNoLt_irref)
∀alpha, ∀p : setprop, ¬ PNoLt alpha p alpha p
Proof:
The rest of this subproof is missing.
Theorem. (PNoLt_trichotomy_or)
∀alpha beta, ∀p q : setprop, ordinal alphaordinal betaPNoLt alpha p beta q alpha = beta PNoEq_ alpha p q PNoLt beta q alpha p
Proof:
The rest of this subproof is missing.
Theorem. (PNoLtEq_tra)
∀alpha beta, ordinal alphaordinal beta∀p q r : setprop, PNoLt alpha p beta qPNoEq_ beta q rPNoLt alpha p beta r
Proof:
The rest of this subproof is missing.
Theorem. (PNoEqLt_tra)
∀alpha beta, ordinal alphaordinal beta∀p q r : setprop, PNoEq_ alpha p qPNoLt alpha q beta rPNoLt alpha p beta r
Proof:
The rest of this subproof is missing.
Theorem. (PNoLt_tra)
∀alpha beta gamma, ordinal alphaordinal betaordinal gamma∀p q r : setprop, PNoLt alpha p beta qPNoLt beta q gamma rPNoLt alpha p gamma r
Proof:
The rest of this subproof is missing.
Definition. We define PNoLe to be λalpha p beta q ⇒ PNoLt alpha p beta q alpha = beta PNoEq_ alpha p q of type set(setprop)set(setprop)prop.
Theorem. (PNoLeI1)
∀alpha beta, ∀p q : setprop, PNoLt alpha p beta qPNoLe alpha p beta q
Proof:
The rest of this subproof is missing.
Theorem. (PNoLeI2)
∀alpha, ∀p q : setprop, PNoEq_ alpha p qPNoLe alpha p alpha q
Proof:
The rest of this subproof is missing.
Theorem. (PNoLe_ref)
∀alpha, ∀p : setprop, PNoLe alpha p alpha p
Proof:
The rest of this subproof is missing.
Theorem. (PNoLe_antisym)
∀alpha beta, ordinal alphaordinal beta∀p q : setprop, PNoLe alpha p beta qPNoLe beta q alpha palpha = beta PNoEq_ alpha p q
Proof:
The rest of this subproof is missing.
Theorem. (PNoLtLe_tra)
∀alpha beta gamma, ordinal alphaordinal betaordinal gamma∀p q r : setprop, PNoLt alpha p beta qPNoLe beta q gamma rPNoLt alpha p gamma r
Proof:
The rest of this subproof is missing.
Theorem. (PNoLeLt_tra)
∀alpha beta gamma, ordinal alphaordinal betaordinal gamma∀p q r : setprop, PNoLe alpha p beta qPNoLt beta q gamma rPNoLt alpha p gamma r
Proof:
The rest of this subproof is missing.
Theorem. (PNoEqLe_tra)
∀alpha beta, ordinal alphaordinal beta∀p q r : setprop, PNoEq_ alpha p qPNoLe alpha q beta rPNoLe alpha p beta r
Proof:
The rest of this subproof is missing.
Theorem. (PNoLe_tra)
∀alpha beta gamma, ordinal alphaordinal betaordinal gamma∀p q r : setprop, PNoLe alpha p beta qPNoLe beta q gamma rPNoLe alpha p gamma r
Proof:
The rest of this subproof is missing.
Definition. We define PNo_downc to be λL alpha p ⇒ ∃beta, ordinal beta ∃q : setprop, L beta q PNoLe alpha p beta q of type (set(setprop)prop)set(setprop)prop.
Definition. We define PNo_upc to be λR alpha p ⇒ ∃beta, ordinal beta ∃q : setprop, R beta q PNoLe beta q alpha p of type (set(setprop)prop)set(setprop)prop.
Theorem. (PNoLe_downc)
∀L : set(setprop)prop, ∀alpha beta, ∀p q : setprop, ordinal alphaordinal betaPNo_downc L alpha pPNoLe beta q alpha pPNo_downc L beta q
Proof:
The rest of this subproof is missing.
Theorem. (PNo_downc_ref)
∀L : set(setprop)prop, ∀alpha, ordinal alpha∀p : setprop, L alpha pPNo_downc L alpha p
Proof:
The rest of this subproof is missing.
Theorem. (PNo_upc_ref)
∀R : set(setprop)prop, ∀alpha, ordinal alpha∀p : setprop, R alpha pPNo_upc R alpha p
Proof:
The rest of this subproof is missing.
Theorem. (PNoLe_upc)
∀R : set(setprop)prop, ∀alpha beta, ∀p q : setprop, ordinal alphaordinal betaPNo_upc R alpha pPNoLe alpha p beta qPNo_upc R beta q
Proof:
The rest of this subproof is missing.
Definition. We define PNoLt_pwise to be λL R ⇒ ∀gamma, ordinal gamma∀p : setprop, L gamma p∀delta, ordinal delta∀q : setprop, R delta qPNoLt gamma p delta q of type (set(setprop)prop)(set(setprop)prop)prop.
Theorem. (PNoLt_pwise_downc_upc)
∀L R : set(setprop)prop, PNoLt_pwise L RPNoLt_pwise (PNo_downc L) (PNo_upc R)
Proof:
The rest of this subproof is missing.
Definition. We define PNo_rel_strict_upperbd to be λL alpha p ⇒ ∀betaalpha, ∀q : setprop, PNo_downc L beta qPNoLt beta q alpha p of type (set(setprop)prop)set(setprop)prop.
Definition. We define PNo_rel_strict_lowerbd to be λR alpha p ⇒ ∀betaalpha, ∀q : setprop, PNo_upc R beta qPNoLt alpha p beta q of type (set(setprop)prop)set(setprop)prop.
Definition. We define PNo_rel_strict_imv to be λL R alpha p ⇒ PNo_rel_strict_upperbd L alpha p PNo_rel_strict_lowerbd R alpha p of type (set(setprop)prop)(set(setprop)prop)set(setprop)prop.
Theorem. (PNoEq_rel_strict_upperbd)
∀L : set(setprop)prop, ∀alpha, ordinal alpha∀p q : setprop, PNoEq_ alpha p qPNo_rel_strict_upperbd L alpha pPNo_rel_strict_upperbd L alpha q
Proof:
The rest of this subproof is missing.
Theorem. (PNo_rel_strict_upperbd_antimon)
∀L : set(setprop)prop, ∀alpha, ordinal alpha∀p : setprop, ∀betaalpha, PNo_rel_strict_upperbd L alpha pPNo_rel_strict_upperbd L beta p
Proof:
The rest of this subproof is missing.
Theorem. (PNoEq_rel_strict_lowerbd)
∀R : set(setprop)prop, ∀alpha, ordinal alpha∀p q : setprop, PNoEq_ alpha p qPNo_rel_strict_lowerbd R alpha pPNo_rel_strict_lowerbd R alpha q
Proof:
The rest of this subproof is missing.
Theorem. (PNo_rel_strict_lowerbd_antimon)
∀R : set(setprop)prop, ∀alpha, ordinal alpha∀p : setprop, ∀betaalpha, PNo_rel_strict_lowerbd R alpha pPNo_rel_strict_lowerbd R beta p
Proof:
The rest of this subproof is missing.
Theorem. (PNoEq_rel_strict_imv)
∀L R : set(setprop)prop, ∀alpha, ordinal alpha∀p q : setprop, PNoEq_ alpha p qPNo_rel_strict_imv L R alpha pPNo_rel_strict_imv L R alpha q
Proof:
The rest of this subproof is missing.
Theorem. (PNo_rel_strict_imv_antimon)
∀L R : set(setprop)prop, ∀alpha, ordinal alpha∀p : setprop, ∀betaalpha, PNo_rel_strict_imv L R alpha pPNo_rel_strict_imv L R beta p
Proof:
The rest of this subproof is missing.
Definition. We define PNo_rel_strict_uniq_imv to be λL R alpha p ⇒ PNo_rel_strict_imv L R alpha p ∀q : setprop, PNo_rel_strict_imv L R alpha qPNoEq_ alpha p q of type (set(setprop)prop)(set(setprop)prop)set(setprop)prop.
Definition. We define PNo_rel_strict_split_imv to be λL R alpha p ⇒ PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta delta alpha) PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta delta = alpha) of type (set(setprop)prop)(set(setprop)prop)set(setprop)prop.
Theorem. (PNo_extend0_eq)
∀alpha, ∀p : setprop, PNoEq_ alpha p (λdelta ⇒ p delta delta alpha)
Proof:
The rest of this subproof is missing.
Theorem. (PNo_extend1_eq)
∀alpha, ∀p : setprop, PNoEq_ alpha p (λdelta ⇒ p delta delta = alpha)
Proof:
The rest of this subproof is missing.
Theorem. (PNo_rel_imv_ex)
∀L R : set(setprop)prop, PNoLt_pwise L R∀alpha, ordinal alpha(∃p : setprop, PNo_rel_strict_uniq_imv L R alpha p) (∃taualpha, ∃p : setprop, PNo_rel_strict_split_imv L R tau p)
Proof:
The rest of this subproof is missing.
Definition. We define PNo_lenbdd to be λalpha L ⇒ ∀beta, ∀p : setprop, L beta pbeta alpha of type set(set(setprop)prop)prop.
Theorem. (PNo_lenbdd_strict_imv_extend0)
∀L R : set(setprop)prop, ∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha R∀p : setprop, PNo_rel_strict_imv L R alpha pPNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta delta alpha)
Proof:
The rest of this subproof is missing.
Theorem. (PNo_lenbdd_strict_imv_extend1)
∀L R : set(setprop)prop, ∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha R∀p : setprop, PNo_rel_strict_imv L R alpha pPNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta delta = alpha)
Proof:
The rest of this subproof is missing.
Theorem. (PNo_lenbdd_strict_imv_split)
∀L R : set(setprop)prop, ∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha R∀p : setprop, PNo_rel_strict_imv L R alpha pPNo_rel_strict_split_imv L R alpha p
Proof:
The rest of this subproof is missing.
Theorem. (PNo_rel_imv_bdd_ex)
∀L R : set(setprop)prop, PNoLt_pwise L R∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha R∃betaordsucc alpha, ∃p : setprop, PNo_rel_strict_split_imv L R beta p
Proof:
The rest of this subproof is missing.
Definition. We define PNo_strict_upperbd to be λL alpha p ⇒ ∀beta, ordinal beta∀q : setprop, L beta qPNoLt beta q alpha p of type (set(setprop)prop)set(setprop)prop.
Definition. We define PNo_strict_lowerbd to be λR alpha p ⇒ ∀beta, ordinal beta∀q : setprop, R beta qPNoLt alpha p beta q of type (set(setprop)prop)set(setprop)prop.
Definition. We define PNo_strict_imv to be λL R alpha p ⇒ PNo_strict_upperbd L alpha p PNo_strict_lowerbd R alpha p of type (set(setprop)prop)(set(setprop)prop)set(setprop)prop.
Theorem. (PNoEq_strict_upperbd)
∀L : set(setprop)prop, ∀alpha, ordinal alpha∀p q : setprop, PNoEq_ alpha p qPNo_strict_upperbd L alpha pPNo_strict_upperbd L alpha q
Proof:
The rest of this subproof is missing.
Theorem. (PNoEq_strict_lowerbd)
∀R : set(setprop)prop, ∀alpha, ordinal alpha∀p q : setprop, PNoEq_ alpha p qPNo_strict_lowerbd R alpha pPNo_strict_lowerbd R alpha q
Proof:
The rest of this subproof is missing.
Theorem. (PNoEq_strict_imv)
∀L R : set(setprop)prop, ∀alpha, ordinal alpha∀p q : setprop, PNoEq_ alpha p qPNo_strict_imv L R alpha pPNo_strict_imv L R alpha q
Proof:
The rest of this subproof is missing.
Theorem. (PNo_strict_upperbd_imp_rel_strict_upperbd)
∀L : set(setprop)prop, ∀alpha, ordinal alpha∀betaordsucc alpha, ∀p : setprop, PNo_strict_upperbd L alpha pPNo_rel_strict_upperbd L beta p
Proof:
The rest of this subproof is missing.
Theorem. (PNo_strict_lowerbd_imp_rel_strict_lowerbd)
∀R : set(setprop)prop, ∀alpha, ordinal alpha∀betaordsucc alpha, ∀p : setprop, PNo_strict_lowerbd R alpha pPNo_rel_strict_lowerbd R beta p
Proof:
The rest of this subproof is missing.
Theorem. (PNo_strict_imv_imp_rel_strict_imv)
∀L R : set(setprop)prop, ∀alpha, ordinal alpha∀betaordsucc alpha, ∀p : setprop, PNo_strict_imv L R alpha pPNo_rel_strict_imv L R beta p
Proof:
The rest of this subproof is missing.
Theorem. (PNo_rel_split_imv_imp_strict_imv)
∀L R : set(setprop)prop, ∀alpha, ordinal alpha∀p : setprop, PNo_rel_strict_split_imv L R alpha pPNo_strict_imv L R alpha p
Proof:
The rest of this subproof is missing.
Theorem. (PNo_lenbdd_strict_imv_ex)
∀L R : set(setprop)prop, PNoLt_pwise L R∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha R∃betaordsucc alpha, ∃p : setprop, PNo_strict_imv L R beta p
Proof:
The rest of this subproof is missing.
Definition. We define PNo_least_rep to be λL R beta p ⇒ ordinal beta PNo_strict_imv L R beta p ∀gammabeta, ∀q : setprop, ¬ PNo_strict_imv L R gamma q of type (set(setprop)prop)(set(setprop)prop)set(setprop)prop.
Definition. We define PNo_least_rep2 to be λL R beta p ⇒ PNo_least_rep L R beta p ∀x, x beta¬ p x of type (set(setprop)prop)(set(setprop)prop)set(setprop)prop.
Theorem. (PNo_strict_imv_pred_eq)
∀L R : set(setprop)prop, PNoLt_pwise L R∀alpha, ordinal alpha∀p q : setprop, PNo_least_rep L R alpha pPNo_strict_imv L R alpha q∀betaalpha, p beta q beta
Proof:
The rest of this subproof is missing.
Theorem. (PNo_lenbdd_least_rep2_exuniq2)
∀L R : set(setprop)prop, PNoLt_pwise L R∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha R∃beta, (∃p : setprop, PNo_least_rep2 L R beta p) (∀p q : setprop, PNo_least_rep2 L R beta pPNo_least_rep2 L R beta qp = q)
Proof:
The rest of this subproof is missing.
Definition. We define PNo_bd to be λL R ⇒ DescrR_i_io_1 (PNo_least_rep2 L R) of type (set(setprop)prop)(set(setprop)prop)set.
Definition. We define PNo_pred to be λL R ⇒ DescrR_i_io_2 (PNo_least_rep2 L R) of type (set(setprop)prop)(set(setprop)prop)setprop.
Theorem. (PNo_bd_pred_lem)
∀L R : set(setprop)prop, PNoLt_pwise L R∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha RPNo_least_rep2 L R (PNo_bd L R) (PNo_pred L R)
Proof:
The rest of this subproof is missing.
Theorem. (PNo_bd_pred)
∀L R : set(setprop)prop, PNoLt_pwise L R∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha RPNo_least_rep L R (PNo_bd L R) (PNo_pred L R)
Proof:
The rest of this subproof is missing.
Theorem. (PNo_bd_In)
∀L R : set(setprop)prop, PNoLt_pwise L R∀alpha, ordinal alphaPNo_lenbdd alpha LPNo_lenbdd alpha RPNo_bd L R ordsucc alpha
Proof:
The rest of this subproof is missing.
Beginning of Section TaggedSets
Let tag : setsetλalpha ⇒ SetAdjoin alpha {1}
Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (tagged_notin_ordinal)
∀alpha y, ordinal alpha(y ') alpha
Proof:
The rest of this subproof is missing.
Theorem. (tagged_eqE_Subq)
∀alpha beta, ordinal alphaalpha ' = beta 'alpha beta
Proof:
The rest of this subproof is missing.
Theorem. (tagged_eqE_eq)
∀alpha beta, ordinal alphaordinal betaalpha ' = beta 'alpha = beta
Proof:
The rest of this subproof is missing.
Theorem. (tagged_ReplE)
∀alpha beta, ordinal alphaordinal betabeta ' {gamma '|gammaalpha}beta alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_notin_tagged_Repl)
∀alpha Y, ordinal alphaalpha {y '|yY}
Proof:
The rest of this subproof is missing.
Definition. We define SNoElts_ to be λalpha ⇒ alpha {beta '|betaalpha} of type setset.
Theorem. (SNoElts_mon)
∀alpha beta, alpha betaSNoElts_ alpha SNoElts_ beta
Proof:
The rest of this subproof is missing.
Definition. We define SNo_ to be λalpha x ⇒ x SNoElts_ alpha ∀betaalpha, exactly1of2 (beta ' x) (beta x) of type setsetprop.
Definition. We define PSNo to be λalpha p ⇒ {betaalpha|p beta} {beta '|betaalpha, ¬ p beta} of type set(setprop)set.
Theorem. (PNoEq_PSNo)
∀alpha, ordinal alpha∀p : setprop, PNoEq_ alpha (λbeta ⇒ beta PSNo alpha p) p
Proof:
The rest of this subproof is missing.
Theorem. (SNo_PSNo)
∀alpha, ordinal alpha∀p : setprop, SNo_ alpha (PSNo alpha p)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_PSNo_eta_)
∀alpha x, ordinal alphaSNo_ alpha xx = PSNo alpha (λbeta ⇒ beta x)
Proof:
The rest of this subproof is missing.
Definition. We define SNo to be λx ⇒ ∃alpha, ordinal alpha SNo_ alpha x of type setprop.
Theorem. (SNo_SNo)
∀alpha, ordinal alpha∀z, SNo_ alpha zSNo z
Proof:
The rest of this subproof is missing.
Definition. We define SNoLev to be λx ⇒ Eps_i (λalpha ⇒ ordinal alpha SNo_ alpha x) of type setset.
Theorem. (SNoLev_uniq_Subq)
∀x alpha beta, ordinal alphaordinal betaSNo_ alpha xSNo_ beta xalpha beta
Proof:
The rest of this subproof is missing.
Theorem. (SNoLev_uniq)
∀x alpha beta, ordinal alphaordinal betaSNo_ alpha xSNo_ beta xalpha = beta
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNoLev_)
∀x, SNo xSNo_ (SNoLev x) x
Proof:
The rest of this subproof is missing.
Theorem. (SNo_PSNo_eta)
∀x, SNo xx = PSNo (SNoLev x) (λbeta ⇒ beta x)
Proof:
The rest of this subproof is missing.
Theorem. (SNoLev_PSNo)
∀alpha, ordinal alpha∀p : setprop, SNoLev (PSNo alpha p) = alpha
Proof:
The rest of this subproof is missing.
Theorem. (SNo_Subq)
∀x y, SNo xSNo ySNoLev x SNoLev y(∀alphaSNoLev x, alpha x alpha y)x y
Proof:
The rest of this subproof is missing.
Definition. We define SNoEq_ to be λalpha x y ⇒ PNoEq_ alpha (λbeta ⇒ beta x) (λbeta ⇒ beta y) of type setsetsetprop.
Theorem. (SNoEq_I)
∀alpha x y, (∀betaalpha, beta x beta y)SNoEq_ alpha x y
Proof:
The rest of this subproof is missing.
Theorem. (SNo_eq)
∀x y, SNo xSNo ySNoLev x = SNoLev ySNoEq_ (SNoLev x) x yx = y
Proof:
The rest of this subproof is missing.
End of Section TaggedSets
Definition. We define SNoLt to be λx y ⇒ PNoLt (SNoLev x) (λbeta ⇒ beta x) (SNoLev y) (λbeta ⇒ beta y) of type setsetprop.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Definition. We define SNoLe to be λx y ⇒ PNoLe (SNoLev x) (λbeta ⇒ beta x) (SNoLev y) (λbeta ⇒ beta y) of type setsetprop.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Theorem. (SNoLtLe)
∀x y, x < yx y
Proof:
The rest of this subproof is missing.
Theorem. (SNoLeE)
∀x y, SNo xSNo yx yx < y x = y
Proof:
The rest of this subproof is missing.
Theorem. (SNoEq_sym_)
∀alpha x y, SNoEq_ alpha x ySNoEq_ alpha y x
Proof:
The rest of this subproof is missing.
Theorem. (SNoEq_tra_)
∀alpha x y z, SNoEq_ alpha x ySNoEq_ alpha y zSNoEq_ alpha x z
Proof:
The rest of this subproof is missing.
Theorem. (SNoLtE)
∀x y, SNo xSNo yx < y∀p : prop, (∀z, SNo zSNoLev z SNoLev x SNoLev ySNoEq_ (SNoLev z) z xSNoEq_ (SNoLev z) z yx < zz < ySNoLev z xSNoLev z yp)(SNoLev x SNoLev ySNoEq_ (SNoLev x) x ySNoLev x yp)(SNoLev y SNoLev xSNoEq_ (SNoLev y) x ySNoLev y xp)p
Proof:
The rest of this subproof is missing.
Theorem. (SNoLtI2)
∀x y, SNoLev x SNoLev ySNoEq_ (SNoLev x) x ySNoLev x yx < y
Proof:
The rest of this subproof is missing.
Theorem. (SNoLtI3)
∀x y, SNoLev y SNoLev xSNoEq_ (SNoLev y) x ySNoLev y xx < y
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_irref)
∀x, ¬ SNoLt x x
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_trichotomy_or)
∀x y, SNo xSNo yx < y x = y y < x
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_trichotomy_or_impred)
∀x y, SNo xSNo y∀p : prop, (x < yp)(x = yp)(y < xp)p
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_tra)
∀x y z, SNo xSNo ySNo zx < yy < zx < z
Proof:
The rest of this subproof is missing.
Theorem. (SNoLe_ref)
∀x, SNoLe x x
Proof:
The rest of this subproof is missing.
Theorem. (SNoLe_antisym)
∀x y, SNo xSNo yx yy xx = y
Proof:
The rest of this subproof is missing.
Theorem. (SNoLtLe_tra)
∀x y z, SNo xSNo ySNo zx < yy zx < z
Proof:
The rest of this subproof is missing.
Theorem. (SNoLeLt_tra)
∀x y z, SNo xSNo ySNo zx yy < zx < z
Proof:
The rest of this subproof is missing.
Theorem. (SNoLe_tra)
∀x y z, SNo xSNo ySNo zx yy zx z
Proof:
The rest of this subproof is missing.
Theorem. (SNoLtLe_or)
∀x y, SNo xSNo yx < y y x
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_PSNo_PNoLt)
∀alpha beta, ∀p q : setprop, ordinal alphaordinal betaPSNo alpha p < PSNo beta qPNoLt alpha p beta q
Proof:
The rest of this subproof is missing.
Theorem. (PNoLt_SNoLt_PSNo)
∀alpha beta, ∀p q : setprop, ordinal alphaordinal betaPNoLt alpha p beta qPSNo alpha p < PSNo beta q
Proof:
The rest of this subproof is missing.
Definition. We define SNoCut to be λL R ⇒ PSNo (PNo_bd (λalpha p ⇒ ordinal alpha PSNo alpha p L) (λalpha p ⇒ ordinal alpha PSNo alpha p R)) (PNo_pred (λalpha p ⇒ ordinal alpha PSNo alpha p L) (λalpha p ⇒ ordinal alpha PSNo alpha p R)) of type setsetset.
Definition. We define SNoCutP to be λL R ⇒ (∀xL, SNo x) (∀yR, SNo y) (∀xL, ∀yR, x < y) of type setsetprop.
Theorem. (SNoCutP_SNoCut)
∀L R, SNoCutP L RSNo (SNoCut L R) SNoLev (SNoCut L R) ordsucc ((xLordsucc (SNoLev x)) (yRordsucc (SNoLev y))) (∀xL, x < SNoCut L R) (∀yR, SNoCut L R < y) (∀z, SNo z(∀xL, x < z)(∀yR, z < y)SNoLev (SNoCut L R) SNoLev z SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) z)
Proof:
The rest of this subproof is missing.
Theorem. (SNoCutP_SNoCut_impred)
∀L R, SNoCutP L R∀p : prop, (SNo (SNoCut L R)SNoLev (SNoCut L R) ordsucc ((xLordsucc (SNoLev x)) (yRordsucc (SNoLev y)))(∀xL, x < SNoCut L R)(∀yR, SNoCut L R < y)(∀z, SNo z(∀xL, x < z)(∀yR, z < y)SNoLev (SNoCut L R) SNoLev z SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) z)p)p
Proof:
The rest of this subproof is missing.
Theorem. (SNoCutP_L_0)
∀L, (∀xL, SNo x)SNoCutP L 0
Proof:
The rest of this subproof is missing.
Theorem. (SNoCutP_0_0)
Proof:
The rest of this subproof is missing.
Definition. We define SNoS_ to be λalpha ⇒ {x𝒫 (SNoElts_ alpha)|∃betaalpha, SNo_ beta x} of type setset.
Theorem. (SNoS_E)
∀alpha, ordinal alpha∀xSNoS_ alpha, ∃betaalpha, SNo_ beta x
Proof:
The rest of this subproof is missing.
Beginning of Section TaggedSets2
Let tag : setsetλalpha ⇒ SetAdjoin alpha {1}
Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag.
Theorem. (SNoS_I)
∀alpha, ordinal alpha∀x, ∀betaalpha, SNo_ beta xx SNoS_ alpha
Proof:
The rest of this subproof is missing.
Theorem. (SNoS_I2)
∀x y, SNo xSNo ySNoLev x SNoLev yx SNoS_ (SNoLev y)
Proof:
The rest of this subproof is missing.
Theorem. (SNoS_Subq)
∀alpha beta, ordinal alphaordinal betaalpha betaSNoS_ alpha SNoS_ beta
Proof:
The rest of this subproof is missing.
Theorem. (SNoLev_uniq2)
∀alpha, ordinal alpha∀x, SNo_ alpha xSNoLev x = alpha
Proof:
The rest of this subproof is missing.
Theorem. (SNoS_E2)
∀alpha, ordinal alpha∀xSNoS_ alpha, ∀p : prop, (SNoLev x alphaordinal (SNoLev x)SNo xSNo_ (SNoLev x) xp)p
Proof:
The rest of this subproof is missing.
Theorem. (SNoS_In_neq)
∀w, SNo w∀xSNoS_ (SNoLev w), x w
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define SNoL to be λz ⇒ {xSNoS_ (SNoLev z)|x < z} of type setset.
Definition. We define SNoR to be λz ⇒ {ySNoS_ (SNoLev z)|z < y} of type setset.
Proof:
The rest of this subproof is missing.
Theorem. (SNoL_E)
∀x, SNo x∀wSNoL x, ∀p : prop, (SNo wSNoLev w SNoLev xw < xp)p
Proof:
The rest of this subproof is missing.
Theorem. (SNoR_E)
∀x, SNo x∀zSNoR x, ∀p : prop, (SNo zSNoLev z SNoLev xx < zp)p
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNoL_SNoS)
∀x, SNo x∀wSNoL x, w SNoS_ (SNoLev x)
Proof:
The rest of this subproof is missing.
Theorem. (SNoR_SNoS)
∀x, SNo x∀zSNoR x, z SNoS_ (SNoLev x)
Proof:
The rest of this subproof is missing.
Theorem. (SNoL_I)
∀z, SNo z∀x, SNo xSNoLev x SNoLev zx < zx SNoL z
Proof:
The rest of this subproof is missing.
Theorem. (SNoR_I)
∀z, SNo z∀y, SNo ySNoLev y SNoLev zz < yy SNoR z
Proof:
The rest of this subproof is missing.
Theorem. (SNo_eta)
∀z, SNo zz = SNoCut (SNoL z) (SNoR z)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNoCutP_SNoCut_L)
∀L R, SNoCutP L R∀xL, x < SNoCut L R
Proof:
The rest of this subproof is missing.
Theorem. (SNoCutP_SNoCut_R)
∀L R, SNoCutP L R∀yR, SNoCut L R < y
Proof:
The rest of this subproof is missing.
Theorem. (SNoCutP_SNoCut_fst)
∀L R, SNoCutP L R∀z, SNo z(∀xL, x < z)(∀yR, z < y)SNoLev (SNoCut L R) SNoLev z SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) z
Proof:
The rest of this subproof is missing.
Theorem. (SNoCut_Le)
∀L1 R1 L2 R2, SNoCutP L1 R1SNoCutP L2 R2(∀wL1, w < SNoCut L2 R2)(∀zR2, SNoCut L1 R1 < z)SNoCut L1 R1 SNoCut L2 R2
Proof:
The rest of this subproof is missing.
Theorem. (SNoCut_ext)
∀L1 R1 L2 R2, SNoCutP L1 R1SNoCutP L2 R2(∀wL1, w < SNoCut L2 R2)(∀zR1, SNoCut L2 R2 < z)(∀wL2, w < SNoCut L1 R1)(∀zR2, SNoCut L1 R1 < z)SNoCut L1 R1 = SNoCut L2 R2
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_SNoL_or_SNoR_impred)
∀x y, SNo xSNo yx < y∀p : prop, (∀zSNoL y, z SNoR xp)(x SNoL yp)(y SNoR xp)p
Proof:
The rest of this subproof is missing.
Theorem. (SNoL_or_SNoR_impred)
∀x y, SNo xSNo y∀p : prop, (x = yp)(∀zSNoL y, z SNoR xp)(x SNoL yp)(y SNoR xp)(∀zSNoR y, z SNoL xp)(x SNoR yp)(y SNoL xp)p
Proof:
The rest of this subproof is missing.
Theorem. (SNoL_SNoCutP_ex)
∀L R, SNoCutP L R∀wSNoL (SNoCut L R), ∃w'L, w w'
Proof:
The rest of this subproof is missing.
Theorem. (SNoR_SNoCutP_ex)
∀L R, SNoCutP L R∀zSNoR (SNoCut L R), ∃z'R, z' z
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_SNo_)
∀alpha, ordinal alphaSNo_ alpha alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_SNo)
∀alpha, ordinal alphaSNo alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_SNoLev)
∀alpha, ordinal alphaSNoLev alpha = alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_SNoLev_max)
∀alpha, ordinal alpha∀z, SNo zSNoLev z alphaz < alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_SNoL)
∀alpha, ordinal alphaSNoL alpha = SNoS_ alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_SNoR)
∀alpha, ordinal alphaSNoR alpha = Empty
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_In_SNoLt)
∀alpha, ordinal alpha∀betaalpha, beta < alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_SNoLev_max_2)
∀alpha, ordinal alpha∀z, SNo zSNoLev z ordsucc alphaz alpha
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_Subq_SNoLe)
∀alpha beta, ordinal alphaordinal betaalpha betaalpha beta
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_SNoLt_In)
∀alpha beta, ordinal alphaordinal betaalpha < betaalpha beta
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNo_max_SNoLev)
∀x, SNo x(∀ySNoS_ (SNoLev x), y < x)SNoLev x = x
Proof:
The rest of this subproof is missing.
Theorem. (SNo_max_ordinal)
∀x, SNo x(∀ySNoS_ (SNoLev x), y < x)ordinal x
Proof:
The rest of this subproof is missing.
Theorem. (pos_low_eq_one)
∀x, SNo x0 < xSNoLev x 1x = 1
Proof:
The rest of this subproof is missing.
Definition. We define SNo_extend0 to be λx ⇒ PSNo (ordsucc (SNoLev x)) (λdelta ⇒ delta x delta SNoLev x) of type setset.
Definition. We define SNo_extend1 to be λx ⇒ PSNo (ordsucc (SNoLev x)) (λdelta ⇒ delta x delta = SNoLev x) of type setset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNoLev_0_eq_0)
∀x, SNo xSNoLev x = 0x = 0
Proof:
The rest of this subproof is missing.
Definition. We define eps_ to be λn ⇒ {0} {(ordsucc m) '|mn} of type setset.
Theorem. (eps_ordinal_In_eq_0)
∀n alpha, ordinal alphaalpha eps_ nalpha = 0
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNo_pos_eps_Lt)
∀n, nat_p n∀xSNoS_ (ordsucc n), 0 < xeps_ n < x
Proof:
The rest of this subproof is missing.
Theorem. (SNo_pos_eps_Le)
∀n, nat_p n∀xSNoS_ (ordsucc (ordsucc n)), 0 < xeps_ n x
Proof:
The rest of this subproof is missing.
Theorem. (eps_SNo_eq)
∀n, nat_p n∀xSNoS_ (ordsucc n), 0 < xSNoEq_ (SNoLev x) (eps_ n) x∃mn, x = eps_ m
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section TaggedSets2
Theorem. (SNo_etaE)
∀z, SNo z∀p : prop, (∀L R, SNoCutP L R(∀xL, SNoLev x SNoLev z)(∀yR, SNoLev y SNoLev z)z = SNoCut L Rp)p
Proof:
The rest of this subproof is missing.
Theorem. (SNo_ind)
∀P : setprop, (∀L R, SNoCutP L R(∀xL, P x)(∀yR, P y)P (SNoCut L R))∀z, SNo zP z
Proof:
The rest of this subproof is missing.
Beginning of Section SurrealRecI
Variable F : set(setset)set
Let default : setEps_i (λ_ ⇒ True)
Let G : set(setsetset)setsetλalpha g ⇒ If_ii (ordinal alpha) (λz : setif z SNoS_ (ordsucc alpha) then F z (λw ⇒ g (SNoLev w) w) else default) (λz : setdefault)
Definition. We define SNo_rec_i to be λz ⇒ In_rec_ii G (SNoLev z) z of type setset.
Hypothesis Fr : ∀z, SNo z∀g h : setset, (∀wSNoS_ (SNoLev z), g w = h w)F z g = F z h
Proof:
The rest of this subproof is missing.
End of Section SurrealRecI
Beginning of Section SurrealRecII
Variable F : set(set(setset))(setset)
Let default : (setset)Descr_ii (λ_ ⇒ True)
Let G : set(setset(setset))set(setset)λalpha g ⇒ If_iii (ordinal alpha) (λz : setIf_ii (z SNoS_ (ordsucc alpha)) (F z (λw ⇒ g (SNoLev w) w)) default) (λz : setdefault)
Definition. We define SNo_rec_ii to be λz ⇒ In_rec_iii G (SNoLev z) z of type set(setset).
Hypothesis Fr : ∀z, SNo z∀g h : set(setset), (∀wSNoS_ (SNoLev z), g w = h w)F z g = F z h
Proof:
The rest of this subproof is missing.
End of Section SurrealRecII
Beginning of Section SurrealRec2
Variable F : setset(setsetset)set
Let G : set(setsetset)set(setset)setλw f z g ⇒ F w z (λx y ⇒ if x = w then g y else f x y)
Let H : set(setsetset)setsetλw f z ⇒ if SNo z then SNo_rec_i (G w f) z else Empty
Definition. We define SNo_rec2 to be SNo_rec_ii H of type setsetset.
Hypothesis Fr : ∀w, SNo w∀z, SNo z∀g h : setsetset, (∀xSNoS_ (SNoLev w), ∀y, SNo yg x y = h x y)(∀ySNoS_ (SNoLev z), g w y = h w y)F w z g = F w z h
Theorem. (SNo_rec2_G_prop)
∀w, SNo w∀f k : setsetset, (∀xSNoS_ (SNoLev w), f x = k x)∀z, SNo z∀g h : setset, (∀uSNoS_ (SNoLev z), g u = h u)G w f z g = G w k z h
Proof:
The rest of this subproof is missing.
Theorem. (SNo_rec2_eq_1)
∀w, SNo w∀f : setsetset, ∀z, SNo zSNo_rec_i (G w f) z = G w f z (SNo_rec_i (G w f))
Proof:
The rest of this subproof is missing.
Theorem. (SNo_rec2_eq)
∀w, SNo w∀z, SNo zSNo_rec2 w z = F w z SNo_rec2
Proof:
The rest of this subproof is missing.
End of Section SurrealRec2
Theorem. (SNo_ordinal_ind)
∀P : setprop, (∀alpha, ordinal alpha∀xSNoS_ alpha, P x)(∀x, SNo xP x)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_ordinal_ind2)
∀P : setsetprop, (∀alpha, ordinal alpha∀beta, ordinal beta∀xSNoS_ alpha, ∀ySNoS_ beta, P x y)(∀x y, SNo xSNo yP x y)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_ordinal_ind3)
∀P : setsetsetprop, (∀alpha, ordinal alpha∀beta, ordinal beta∀gamma, ordinal gamma∀xSNoS_ alpha, ∀ySNoS_ beta, ∀zSNoS_ gamma, P x y z)(∀x y z, SNo xSNo ySNo zP x y z)
Proof:
The rest of this subproof is missing.
Theorem. (SNoLev_ind)
∀P : setprop, (∀x, SNo x(∀wSNoS_ (SNoLev x), P w)P x)(∀x, SNo xP x)
Proof:
The rest of this subproof is missing.
Theorem. (SNoLev_ind2)
∀P : setsetprop, (∀x y, SNo xSNo y(∀wSNoS_ (SNoLev x), P w y)(∀zSNoS_ (SNoLev y), P x z)(∀wSNoS_ (SNoLev x), ∀zSNoS_ (SNoLev y), P w z)P x y)∀x y, SNo xSNo yP x y
Proof:
The rest of this subproof is missing.
Theorem. (SNoLev_ind3)
∀P : setsetsetprop, (∀x y z, SNo xSNo ySNo z(∀uSNoS_ (SNoLev x), P u y z)(∀vSNoS_ (SNoLev y), P x v z)(∀wSNoS_ (SNoLev z), P x y w)(∀uSNoS_ (SNoLev x), ∀vSNoS_ (SNoLev y), P u v z)(∀uSNoS_ (SNoLev x), ∀wSNoS_ (SNoLev z), P u y w)(∀vSNoS_ (SNoLev y), ∀wSNoS_ (SNoLev z), P x v w)(∀uSNoS_ (SNoLev x), ∀vSNoS_ (SNoLev y), ∀wSNoS_ (SNoLev z), P u v w)P x y z)∀x y z, SNo xSNo ySNo zP x y z
Proof:
The rest of this subproof is missing.
Theorem. (SNo_omega)
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_0_1)
0 < 1
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_0_2)
0 < 2
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_1_2)
1 < 2
Proof:
The rest of this subproof is missing.
Theorem. (restr_SNo_)
∀x, SNo x∀alphaSNoLev x, SNo_ alpha (x SNoElts_ alpha)
Proof:
The rest of this subproof is missing.
Theorem. (restr_SNo)
∀x, SNo x∀alphaSNoLev x, SNo (x SNoElts_ alpha)
Proof:
The rest of this subproof is missing.
Theorem. (restr_SNoLev)
∀x, SNo x∀alphaSNoLev x, SNoLev (x SNoElts_ alpha) = alpha
Proof:
The rest of this subproof is missing.
Theorem. (restr_SNoEq)
∀x, SNo x∀alphaSNoLev x, SNoEq_ alpha (x SNoElts_ alpha) x
Proof:
The rest of this subproof is missing.
Theorem. (SNo_extend0_restr_eq)
∀x, SNo xx = SNo_extend0 x SNoElts_ (SNoLev x)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_extend1_restr_eq)
∀x, SNo xx = SNo_extend1 x SNoElts_ (SNoLev x)
Proof:
The rest of this subproof is missing.
Beginning of Section SurrealMinus
Definition. We define minus_SNo to be SNo_rec_i (λx m ⇒ SNoCut {m z|zSNoR x} {m w|wSNoL x}) of type setset.
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_prop1)
∀x, SNo xSNo (- x) (∀uSNoL x, - x < - u) (∀uSNoR x, - u < - x) SNoCutP {- z|zSNoR x} {- w|wSNoL x}
Proof:
The rest of this subproof is missing.
Theorem. (SNo_minus_SNo)
∀x, SNo xSNo (- x)
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_Lt_contra)
∀x y, SNo xSNo yx < y- y < - x
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_Le_contra)
∀x y, SNo xSNo yx y- y - x
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_Lev_lem1)
∀alpha, ordinal alpha∀xSNoS_ alpha, SNoLev (- x) SNoLev x
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_invol)
∀x, SNo x- - x = x
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_SNo_)
∀alpha, ordinal alpha∀x, SNo_ alpha xSNo_ alpha (- x)
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_SNoS_)
∀alpha, ordinal alpha∀x, x SNoS_ alpha- x SNoS_ alpha
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNoCut_eq_lem)
∀v, SNo v∀L R, SNoCutP L Rv = SNoCut L R- v = SNoCut {- z|zR} {- w|wL}
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_Lt_contra1)
∀x y, SNo xSNo y- x < y- y < x
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_Lt_contra2)
∀x y, SNo xSNo yx < - yy < - x
Proof:
The rest of this subproof is missing.
Theorem. (mordinal_SNoLev_min_2)
∀alpha, ordinal alpha∀z, SNo zSNoLev z ordsucc alpha- alpha z
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section SurrealMinus
Beginning of Section SurrealAdd
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Definition. We define add_SNo to be SNo_rec2 (λx y a ⇒ SNoCut ({a w y|wSNoL x} {a x w|wSNoL y}) ({a z y|zSNoR x} {a x z|zSNoR y})) of type setsetset.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Theorem. (add_SNo_eq)
∀x, SNo x∀y, SNo yx + y = SNoCut ({w + y|wSNoL x} {x + w|wSNoL y}) ({z + y|zSNoR x} {x + z|zSNoR y})
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_prop1)
∀x y, SNo xSNo ySNo (x + y) (∀uSNoL x, u + y < x + y) (∀uSNoR x, x + y < u + y) (∀uSNoL y, x + u < x + y) (∀uSNoR y, x + y < x + u) SNoCutP ({w + y|wSNoL x} {x + w|wSNoL y}) ({z + y|zSNoR x} {x + z|zSNoR y})
Proof:
The rest of this subproof is missing.
Theorem. (SNo_add_SNo)
∀x y, SNo xSNo ySNo (x + y)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_add_SNo_3)
∀x y z, SNo xSNo ySNo zSNo (x + y + z)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_add_SNo_3c)
∀x y z, SNo xSNo ySNo zSNo (x + y + - z)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_add_SNo_4)
∀x y z w, SNo xSNo ySNo zSNo wSNo (x + y + z + w)
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt1)
∀x y z, SNo xSNo ySNo zx < zx + y < z + y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Le1)
∀x y z, SNo xSNo ySNo zx zx + y z + y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt2)
∀x y z, SNo xSNo ySNo zy < zx + y < x + z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Le2)
∀x y z, SNo xSNo ySNo zy zx + y x + z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt3a)
∀x y z w, SNo xSNo ySNo zSNo wx < zy wx + y < z + w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt3b)
∀x y z w, SNo xSNo ySNo zSNo wx zy < wx + y < z + w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt3)
∀x y z w, SNo xSNo ySNo zSNo wx < zy < wx + y < z + w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Le3)
∀x y z w, SNo xSNo ySNo zSNo wx zy wx + y z + w
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_com)
∀x y, SNo xSNo yx + y = y + x
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_0L)
∀x, SNo x0 + x = x
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_0R)
∀x, SNo xx + 0 = x
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_ordinal_eq)
∀alpha, ordinal alpha∀beta, ordinal betaalpha + beta = SNoCut ({x + beta|xSNoS_ alpha} {alpha + x|xSNoS_ beta}) Empty
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_ordinal_ordinal)
∀alpha, ordinal alpha∀beta, ordinal betaordinal (alpha + beta)
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_ordinal_SL)
∀alpha, ordinal alpha∀beta, ordinal betaordsucc alpha + beta = ordsucc (alpha + beta)
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_ordinal_SR)
∀alpha, ordinal alpha∀beta, ordinal betaalpha + ordsucc beta = ordsucc (alpha + beta)
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_ordinal_InL)
∀alpha, ordinal alpha∀beta, ordinal beta∀gammaalpha, gamma + beta alpha + beta
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_ordinal_InR)
∀alpha, ordinal alpha∀beta, ordinal beta∀gammabeta, alpha + gamma alpha + beta
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_SNoL_interpolate)
∀x y, SNo xSNo y∀uSNoL (x + y), (∃vSNoL x, u v + y) (∃vSNoL y, u x + v)
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_SNoR_interpolate)
∀x y, SNo xSNo y∀uSNoR (x + y), (∃vSNoR x, v + y u) (∃vSNoR y, x + v u)
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_assoc)
∀x y z, SNo xSNo ySNo zx + (y + z) = (x + y) + z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_R2)
∀x y, SNo xSNo y(x + y) + - y = x
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_R2')
∀x y, SNo xSNo y(x + - y) + y = x
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_L2)
∀x y, SNo xSNo y- x + (x + y) = y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_L2')
∀x y, SNo xSNo yx + (- x + y) = y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_cancel_L)
∀x y z, SNo xSNo ySNo zx + y = x + zy = z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_cancel_R)
∀x y z, SNo xSNo ySNo zx + y = z + yx = z
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (minus_add_SNo_distr)
∀x y, SNo xSNo y- (x + y) = (- x) + (- y)
Proof:
The rest of this subproof is missing.
Theorem. (minus_add_SNo_distr_3)
∀x y z, SNo xSNo ySNo z- (x + y + z) = - x + - y + - z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lev_bd)
∀x y, SNo xSNo ySNoLev (x + y) SNoLev x + SNoLev y
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt1_cancel)
∀x y z, SNo xSNo ySNo zx + y < z + yx < z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt2_cancel)
∀x y z, SNo xSNo ySNo zx + y < x + zy < z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Le1_cancel)
∀x y z, SNo xSNo ySNo zx + y z + yx z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_assoc_4)
∀x y z w, SNo xSNo ySNo zSNo wx + y + z + w = (x + y + z) + w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_com_3_0_1)
∀x y z, SNo xSNo ySNo zx + y + z = y + x + z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_com_3b_1_2)
∀x y z, SNo xSNo ySNo z(x + y) + z = (x + z) + y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_com_4_inner_mid)
∀x y z w, SNo xSNo ySNo zSNo w(x + y) + (z + w) = (x + z) + (y + w)
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_rotate_3_1)
∀x y z, SNo xSNo ySNo zx + y + z = z + x + y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_rotate_4_1)
∀x y z w, SNo xSNo ySNo zSNo wx + y + z + w = w + x + y + z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_rotate_5_1)
∀x y z w v, SNo xSNo ySNo zSNo wSNo vx + y + z + w + v = v + x + y + z + w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_rotate_5_2)
∀x y z w v, SNo xSNo ySNo zSNo wSNo vx + y + z + w + v = w + v + x + y + z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_SNo_prop2)
∀x y, SNo xSNo yx + - x + y = y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_SNo_prop3)
∀x y z w, SNo xSNo ySNo zSNo w(x + y + z) + (- z + w) = x + y + w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_SNo_prop5)
∀x y z w, SNo xSNo ySNo zSNo w(x + y + - z) + (z + w) = x + y + w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Lt1)
∀x y z, SNo xSNo ySNo zx + - y < zx < z + y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Lt2)
∀x y z, SNo xSNo ySNo zz < x + - yz + y < x
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Lt1b)
∀x y z, SNo xSNo ySNo zx < z + yx + - y < z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Lt2b)
∀x y z, SNo xSNo ySNo zz + y < xz < x + - y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Lt1b3)
∀x y z w, SNo xSNo ySNo zSNo wx + y < w + zx + y + - z < w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Lt2b3)
∀x y z w, SNo xSNo ySNo zSNo ww + z < x + yw < x + y + - z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Lt_lem)
∀x y z u v w, SNo xSNo ySNo zSNo uSNo vSNo wx + y + w < u + v + zx + y + - z < u + v + - w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Le2)
∀x y z, SNo xSNo ySNo zz x + - yz + y x
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Le2b)
∀x y z, SNo xSNo ySNo zz + y xz x + - y
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt_subprop2)
∀x y z w u v, SNo xSNo ySNo zSNo wSNo uSNo vx + u < z + vy + v < w + ux + y < z + w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt_subprop3a)
∀x y z w u a, SNo xSNo ySNo zSNo wSNo uSNo ax + z < w + ay + a < ux + y + z < w + u
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt_subprop3b)
∀x y w u v a, SNo xSNo ySNo wSNo uSNo vSNo ax + a < w + vy < a + ux + y < w + u + v
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt_subprop3c)
∀x y z w u a b c, SNo xSNo ySNo zSNo wSNo uSNo aSNo bSNo cx + a < b + cy + c < ub + z < w + ax + y + z < w + u
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt_subprop3d)
∀x y w u v a b c, SNo xSNo ySNo wSNo uSNo vSNo aSNo bSNo cx + a < b + vy < c + ub + c < w + ax + y < w + u + v
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_ordsucc_SNo_eq)
∀alpha, ordinal alphaordsucc alpha = 1 + alpha
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_3a_2b)
∀x y z w u, SNo xSNo ySNo zSNo wSNo u(x + y + z) + (w + u) = (u + y + z) + (w + x)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_eps_Lt)
∀x, SNo x∀nω, x < x + eps_ n
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_eps_Lt')
∀x y, SNo xSNo y∀nω, x < yx < y + eps_ n
Proof:
The rest of this subproof is missing.
Theorem. (SNoLt_minus_pos)
∀x y, SNo xSNo yx < y0 < y + - x
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_omega_In_cases)
∀m, ∀nω, ∀k, nat_p km n + km n m + - n k
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_Lt4)
∀x y z w u v, SNo xSNo ySNo zSNo wSNo uSNo vx < wy < uz < vx + y + z < w + u + v
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_3_3_3_Lt1)
∀x y z w u, SNo xSNo ySNo zSNo wSNo ux + y < z + wx + y + u < z + w + u
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_3_2_3_Lt1)
∀x y z w u, SNo xSNo ySNo zSNo wSNo uy + x < z + wx + u + y < z + w + u
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Lt12b3)
∀x y z w u v, SNo xSNo ySNo zSNo wSNo uSNo vx + y + v < w + u + zx + y + - z < w + u + - v
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Le1b)
∀x y z, SNo xSNo ySNo zx z + yx + - y z
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Le1b3)
∀x y z w, SNo xSNo ySNo zSNo wx + y w + zx + y + - z w
Proof:
The rest of this subproof is missing.
Theorem. (add_SNo_minus_Le12b3)
∀x y z w u v, SNo xSNo ySNo zSNo wSNo uSNo vx + y + v w + u + zx + y + - z w + u + - v
Proof:
The rest of this subproof is missing.
End of Section SurrealAdd
Beginning of Section SurrealAbs
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Definition. We define abs_SNo to be λx ⇒ if 0 x then x else - x of type setset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (pos_abs_SNo)
∀x, 0 < xabs_SNo x = x
Proof:
The rest of this subproof is missing.
Theorem. (neg_abs_SNo)
∀x, SNo xx < 0abs_SNo x = - x
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (abs_SNo_dist_swap)
∀x y, SNo xSNo yabs_SNo (x + - y) = abs_SNo (y + - x)
Proof:
The rest of this subproof is missing.
End of Section SurrealAbs
Beginning of Section SurrealMul
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Definition. We define mul_SNo to be SNo_rec2 (λx y m ⇒ SNoCut ({m (w 0) y + m x (w 1) + - m (w 0) (w 1)|wSNoL x SNoL y} {m (z 0) y + m x (z 1) + - m (z 0) (z 1)|zSNoR x SNoR y}) ({m (w 0) y + m x (w 1) + - m (w 0) (w 1)|wSNoL x SNoR y} {m (z 0) y + m x (z 1) + - m (z 0) (z 1)|zSNoR x SNoL y})) of type setsetset.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Theorem. (mul_SNo_eq)
∀x, SNo x∀y, SNo yx * y = SNoCut ({(w 0) * y + x * (w 1) + - (w 0) * (w 1)|wSNoL x SNoL y} {(z 0) * y + x * (z 1) + - (z 0) * (z 1)|zSNoR x SNoR y}) ({(w 0) * y + x * (w 1) + - (w 0) * (w 1)|wSNoL x SNoR y} {(z 0) * y + x * (z 1) + - (z 0) * (z 1)|zSNoR x SNoL y})
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_eq_2)
∀x y, SNo xSNo y∀p : prop, (∀L R, (∀u, u L(∀q : prop, (∀w0SNoL x, ∀w1SNoL y, u = w0 * y + x * w1 + - w0 * w1q)(∀z0SNoR x, ∀z1SNoR y, u = z0 * y + x * z1 + - z0 * z1q)q))(∀w0SNoL x, ∀w1SNoL y, w0 * y + x * w1 + - w0 * w1 L)(∀z0SNoR x, ∀z1SNoR y, z0 * y + x * z1 + - z0 * z1 L)(∀u, u R(∀q : prop, (∀w0SNoL x, ∀z1SNoR y, u = w0 * y + x * z1 + - w0 * z1q)(∀z0SNoR x, ∀w1SNoL y, u = z0 * y + x * w1 + - z0 * w1q)q))(∀w0SNoL x, ∀z1SNoR y, w0 * y + x * z1 + - w0 * z1 R)(∀z0SNoR x, ∀w1SNoL y, z0 * y + x * w1 + - z0 * w1 R)x * y = SNoCut L Rp)p
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_prop_1)
∀x, SNo x∀y, SNo y∀p : prop, (SNo (x * y)(∀uSNoL x, ∀vSNoL y, u * y + x * v < x * y + u * v)(∀uSNoR x, ∀vSNoR y, u * y + x * v < x * y + u * v)(∀uSNoL x, ∀vSNoR y, x * y + u * v < u * y + x * v)(∀uSNoR x, ∀vSNoL y, x * y + u * v < u * y + x * v)p)p
Proof:
The rest of this subproof is missing.
Theorem. (SNo_mul_SNo)
∀x y, SNo xSNo ySNo (x * y)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_mul_SNo_lem)
∀x y u v, SNo xSNo ySNo uSNo vSNo (u * y + x * v + - (u * v))
Proof:
The rest of this subproof is missing.
Theorem. (SNo_mul_SNo_3)
∀x y z, SNo xSNo ySNo zSNo (x * y * z)
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_eq_3)
∀x y, SNo xSNo y∀p : prop, (∀L R, SNoCutP L R(∀u, u L(∀q : prop, (∀w0SNoL x, ∀w1SNoL y, u = w0 * y + x * w1 + - w0 * w1q)(∀z0SNoR x, ∀z1SNoR y, u = z0 * y + x * z1 + - z0 * z1q)q))(∀w0SNoL x, ∀w1SNoL y, w0 * y + x * w1 + - w0 * w1 L)(∀z0SNoR x, ∀z1SNoR y, z0 * y + x * z1 + - z0 * z1 L)(∀u, u R(∀q : prop, (∀w0SNoL x, ∀z1SNoR y, u = w0 * y + x * z1 + - w0 * z1q)(∀z0SNoR x, ∀w1SNoL y, u = z0 * y + x * w1 + - z0 * w1q)q))(∀w0SNoL x, ∀z1SNoR y, w0 * y + x * z1 + - w0 * z1 R)(∀z0SNoR x, ∀w1SNoL y, z0 * y + x * w1 + - z0 * w1 R)x * y = SNoCut L Rp)p
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_Lt)
∀x y u v, SNo xSNo ySNo uSNo vu < xv < yu * y + x * v < x * y + u * v
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_Le)
∀x y u v, SNo xSNo ySNo uSNo vu xv yu * y + x * v x * y + u * v
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_SNoL_interpolate)
∀x y, SNo xSNo y∀uSNoL (x * y), (∃vSNoL x, ∃wSNoL y, u + v * w v * y + x * w) (∃vSNoR x, ∃wSNoR y, u + v * w v * y + x * w)
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_SNoL_interpolate_impred)
∀x y, SNo xSNo y∀uSNoL (x * y), ∀p : prop, (∀vSNoL x, ∀wSNoL y, u + v * w v * y + x * wp)(∀vSNoR x, ∀wSNoR y, u + v * w v * y + x * wp)p
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_SNoR_interpolate)
∀x y, SNo xSNo y∀uSNoR (x * y), (∃vSNoL x, ∃wSNoR y, v * y + x * w u + v * w) (∃vSNoR x, ∃wSNoL y, v * y + x * w u + v * w)
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_SNoR_interpolate_impred)
∀x y, SNo xSNo y∀uSNoR (x * y), ∀p : prop, (∀vSNoL x, ∀wSNoR y, v * y + x * w u + v * wp)(∀vSNoR x, ∀wSNoL y, v * y + x * w u + v * wp)p
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_Subq_lem)
∀x y X Y Z W, ∀U U', (∀u, u U(∀q : prop, (∀w0X, ∀w1Y, u = w0 * y + x * w1 + - w0 * w1q)(∀z0Z, ∀z1W, u = z0 * y + x * z1 + - z0 * z1q)q))(∀w0X, ∀w1Y, w0 * y + x * w1 + - w0 * w1 U')(∀w0Z, ∀w1W, w0 * y + x * w1 + - w0 * w1 U')U U'
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_zeroR)
∀x, SNo xx * 0 = 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_oneR)
∀x, SNo xx * 1 = x
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_com)
∀x y, SNo xSNo yx * y = y * x
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_minus_distrL)
∀x y, SNo xSNo y(- x) * y = - x * y
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_minus_distrR)
∀x y, SNo xSNo yx * (- y) = - (x * y)
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_distrR)
∀x y z, SNo xSNo ySNo z(x + y) * z = x * z + y * z
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_distrL)
∀x y z, SNo xSNo ySNo zx * (y + z) = x * y + x * z
Proof:
The rest of this subproof is missing.
Beginning of Section mul_SNo_assoc_lems
Variable M : setsetset
Hypothesis DL : ∀x y z, SNo xSNo ySNo zx * (y + z) = x * y + x * z
Hypothesis DR : ∀x y z, SNo xSNo ySNo z(x + y) * z = x * z + y * z
Hypothesis IL : ∀x y, SNo xSNo y∀uSNoL (x * y), ∀p : prop, (∀vSNoL x, ∀wSNoL y, u + v * w v * y + x * wp)(∀vSNoR x, ∀wSNoR y, u + v * w v * y + x * wp)p
Hypothesis IR : ∀x y, SNo xSNo y∀uSNoR (x * y), ∀p : prop, (∀vSNoL x, ∀wSNoR y, v * y + x * w u + v * wp)(∀vSNoR x, ∀wSNoL y, v * y + x * w u + v * wp)p
Hypothesis M_Lt : ∀x y u v, SNo xSNo ySNo uSNo vu < xv < yu * y + x * v < x * y + u * v
Hypothesis M_Le : ∀x y u v, SNo xSNo ySNo uSNo vu xv yu * y + x * v x * y + u * v
Theorem. (mul_SNo_assoc_lem1)
∀x y z, SNo xSNo ySNo z(∀uSNoS_ (SNoLev x), u * (y * z) = (u * y) * z)(∀vSNoS_ (SNoLev y), x * (v * z) = (x * v) * z)(∀wSNoS_ (SNoLev z), x * (y * w) = (x * y) * w)(∀uSNoS_ (SNoLev x), ∀vSNoS_ (SNoLev y), u * (v * z) = (u * v) * z)(∀uSNoS_ (SNoLev x), ∀wSNoS_ (SNoLev z), u * (y * w) = (u * y) * w)(∀vSNoS_ (SNoLev y), ∀wSNoS_ (SNoLev z), x * (v * w) = (x * v) * w)(∀uSNoS_ (SNoLev x), ∀vSNoS_ (SNoLev y), ∀wSNoS_ (SNoLev z), u * (v * w) = (u * v) * w)∀L, (∀uL, ∀q : prop, (∀vSNoL x, ∀wSNoL (y * z), u = v * (y * z) + x * w + - v * wq)(∀vSNoR x, ∀wSNoR (y * z), u = v * (y * z) + x * w + - v * wq)q)∀uL, u < (x * y) * z
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_assoc_lem2)
∀x y z, SNo xSNo ySNo z(∀uSNoS_ (SNoLev x), u * (y * z) = (u * y) * z)(∀vSNoS_ (SNoLev y), x * (v * z) = (x * v) * z)(∀wSNoS_ (SNoLev z), x * (y * w) = (x * y) * w)(∀uSNoS_ (SNoLev x), ∀vSNoS_ (SNoLev y), u * (v * z) = (u * v) * z)(∀uSNoS_ (SNoLev x), ∀wSNoS_ (SNoLev z), u * (y * w) = (u * y) * w)(∀vSNoS_ (SNoLev y), ∀wSNoS_ (SNoLev z), x * (v * w) = (x * v) * w)(∀uSNoS_ (SNoLev x), ∀vSNoS_ (SNoLev y), ∀wSNoS_ (SNoLev z), u * (v * w) = (u * v) * w)∀R, (∀uR, ∀q : prop, (∀vSNoL x, ∀wSNoR (y * z), u = v * (y * z) + x * w + - v * wq)(∀vSNoR x, ∀wSNoL (y * z), u = v * (y * z) + x * w + - v * wq)q)∀uR, (x * y) * z < u
Proof:
The rest of this subproof is missing.
End of Section mul_SNo_assoc_lems
Theorem. (mul_SNo_assoc)
∀x y z, SNo xSNo ySNo zx * (y * z) = (x * y) * z
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_zeroL)
∀x, SNo x0 * x = 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_oneL)
∀x, SNo x1 * x = x
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_rotate_3_1)
∀x y z, SNo xSNo ySNo zx * y * z = z * x * y
Proof:
The rest of this subproof is missing.
Theorem. (pos_mul_SNo_Lt)
∀x y z, SNo x0 < xSNo ySNo zy < zx * y < x * z
Proof:
The rest of this subproof is missing.
Theorem. (nonneg_mul_SNo_Le)
∀x y z, SNo x0 xSNo ySNo zy zx * y x * z
Proof:
The rest of this subproof is missing.
Theorem. (neg_mul_SNo_Lt)
∀x y z, SNo xx < 0SNo ySNo zz < yx * y < x * z
Proof:
The rest of this subproof is missing.
Theorem. (pos_mul_SNo_Lt')
∀x y z, SNo xSNo ySNo z0 < zx < yx * z < y * z
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_Lt1_pos_Lt)
∀x y, SNo xSNo yx < 10 < yx * y < y
Proof:
The rest of this subproof is missing.
Theorem. (nonneg_mul_SNo_Le')
∀x y z, SNo xSNo ySNo z0 zx yx * z y * z
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_Le1_nonneg_Le)
∀x y, SNo xSNo yx 10 yx * y y
Proof:
The rest of this subproof is missing.
Theorem. (pos_mul_SNo_Lt2)
∀x y z w, SNo xSNo ySNo zSNo w0 < x0 < yx < zy < wx * y < z * w
Proof:
The rest of this subproof is missing.
Theorem. (nonneg_mul_SNo_Le2)
∀x y z w, SNo xSNo ySNo zSNo w0 x0 yx zy wx * y z * w
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_pos_pos)
∀x y, SNo xSNo y0 < x0 < y0 < x * y
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_pos_neg)
∀x y, SNo xSNo y0 < xy < 0x * y < 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_neg_pos)
∀x y, SNo xSNo yx < 00 < yx * y < 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_neg_neg)
∀x y, SNo xSNo yx < 0y < 00 < x * y
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_nonneg_nonneg)
∀x y, SNo xSNo y0 x0 y0 x * y
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_nonpos_pos)
∀x y, SNo xSNo yx 00 < yx * y 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_nonpos_neg)
∀x y, SNo xSNo yx 0y < 00 x * y
Proof:
The rest of this subproof is missing.
Theorem. (nonpos_mul_SNo_Le)
∀x y z, SNo xx 0SNo ySNo zz yx * y x * z
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNo_pos_sqr_uniq)
∀x y, SNo xSNo y0 < x0 < yx * x = y * yx = y
Proof:
The rest of this subproof is missing.
Theorem. (SNo_nonneg_sqr_uniq)
∀x y, SNo xSNo y0 x0 yx * x = y * yx = y
Proof:
The rest of this subproof is missing.
Theorem. (SNo_foil)
∀x y z w, SNo xSNo ySNo zSNo w(x + y) * (z + w) = x * z + x * w + y * z + y * w
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_minus_minus)
∀x y, SNo xSNo y(- x) * (- y) = x * y
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_com_3_0_1)
∀x y z, SNo xSNo ySNo zx * y * z = y * x * z
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_com_3b_1_2)
∀x y z, SNo xSNo ySNo z(x * y) * z = (x * z) * y
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_com_4_inner_mid)
∀x y z w, SNo xSNo ySNo zSNo w(x * y) * (z * w) = (x * z) * (y * w)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_foil_mm)
∀x y z w, SNo xSNo ySNo zSNo w(x + - y) * (z + - w) = x * z + - x * w + - y * z + y * w
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_nonzero_cancel)
∀x y z, SNo xx 0SNo ySNo zx * y = x * zy = z
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNoCutP_lem)
∀Lx Rx Ly Ry x y, SNoCutP Lx RxSNoCutP Ly Ryx = SNoCut Lx Rxy = SNoCut Ly RySNoCutP ({w 0 * y + x * w 1 + - w 0 * w 1|wLx Ly} {z 0 * y + x * z 1 + - z 0 * z 1|zRx Ry}) ({w 0 * y + x * w 1 + - w 0 * w 1|wLx Ry} {z 0 * y + x * z 1 + - z 0 * z 1|zRx Ly}) x * y = SNoCut ({w 0 * y + x * w 1 + - w 0 * w 1|wLx Ly} {z 0 * y + x * z 1 + - z 0 * z 1|zRx Ry}) ({w 0 * y + x * w 1 + - w 0 * w 1|wLx Ry} {z 0 * y + x * z 1 + - z 0 * z 1|zRx Ly}) ∀q : prop, (∀LxLy' RxRy' LxRy' RxLy', (∀uLxLy', ∀p : prop, (∀wLx, ∀w'Ly, SNo wSNo w'w < xw' < yu = w * y + x * w' + - w * w'p)p)(∀wLx, ∀w'Ly, w * y + x * w' + - w * w' LxLy')(∀uRxRy', ∀p : prop, (∀zRx, ∀z'Ry, SNo zSNo z'x < zy < z'u = z * y + x * z' + - z * z'p)p)(∀zRx, ∀z'Ry, z * y + x * z' + - z * z' RxRy')(∀uLxRy', ∀p : prop, (∀wLx, ∀zRy, SNo wSNo zw < xy < zu = w * y + x * z + - w * zp)p)(∀wLx, ∀zRy, w * y + x * z + - w * z LxRy')(∀uRxLy', ∀p : prop, (∀zRx, ∀wLy, SNo zSNo wx < zw < yu = z * y + x * w + - z * wp)p)(∀zRx, ∀wLy, z * y + x * w + - z * w RxLy')SNoCutP (LxLy' RxRy') (LxRy' RxLy')x * y = SNoCut (LxLy' RxRy') (LxRy' RxLy')q)q
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNoCut_abs)
∀Lx Rx Ly Ry x y, SNoCutP Lx RxSNoCutP Ly Ryx = SNoCut Lx Rxy = SNoCut Ly Ry∀q : prop, (∀LxLy' RxRy' LxRy' RxLy', (∀uLxLy', ∀p : prop, (∀wLx, ∀w'Ly, SNo wSNo w'w < xw' < yu = w * y + x * w' + - w * w'p)p)(∀wLx, ∀w'Ly, w * y + x * w' + - w * w' LxLy')(∀uRxRy', ∀p : prop, (∀zRx, ∀z'Ry, SNo zSNo z'x < zy < z'u = z * y + x * z' + - z * z'p)p)(∀zRx, ∀z'Ry, z * y + x * z' + - z * z' RxRy')(∀uLxRy', ∀p : prop, (∀wLx, ∀zRy, SNo wSNo zw < xy < zu = w * y + x * z + - w * zp)p)(∀wLx, ∀zRy, w * y + x * z + - w * z LxRy')(∀uRxLy', ∀p : prop, (∀zRx, ∀wLy, SNo zSNo wx < zw < yu = z * y + x * w + - z * wp)p)(∀zRx, ∀wLy, z * y + x * w + - z * w RxLy')SNoCutP (LxLy' RxRy') (LxRy' RxLy')x * y = SNoCut (LxLy' RxRy') (LxRy' RxLy')q)q
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_SNoCut_SNoL_interpolate)
∀Lx Rx Ly Ry, SNoCutP Lx RxSNoCutP Ly Ry∀x y, x = SNoCut Lx Rxy = SNoCut Ly Ry∀uSNoL (x * y), (∃vLx, ∃wLy, u + v * w v * y + x * w) (∃vRx, ∃wRy, u + v * w v * y + x * w)
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_SNoCut_SNoL_interpolate_impred)
∀Lx Rx Ly Ry, SNoCutP Lx RxSNoCutP Ly Ry∀x y, x = SNoCut Lx Rxy = SNoCut Ly Ry∀uSNoL (x * y), ∀p : prop, (∀vLx, ∀wLy, u + v * w v * y + x * wp)(∀vRx, ∀wRy, u + v * w v * y + x * wp)p
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_SNoCut_SNoR_interpolate)
∀Lx Rx Ly Ry, SNoCutP Lx RxSNoCutP Ly Ry∀x y, x = SNoCut Lx Rxy = SNoCut Ly Ry∀uSNoR (x * y), (∃vLx, ∃wRy, v * y + x * w u + v * w) (∃vRx, ∃wLy, v * y + x * w u + v * w)
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_SNoCut_SNoR_interpolate_impred)
∀Lx Rx Ly Ry, SNoCutP Lx RxSNoCutP Ly Ry∀x y, x = SNoCut Lx Rxy = SNoCut Ly Ry∀uSNoR (x * y), ∀p : prop, (∀vLx, ∀wRy, v * y + x * w u + v * wp)(∀vRx, ∀wLy, v * y + x * w u + v * wp)p
Proof:
The rest of this subproof is missing.
Theorem. (nonpos_nonneg_0)
∀m nω, m = - nm = 0 n = 0
Proof:
The rest of this subproof is missing.
Theorem. (mul_minus_SNo_distrR)
∀x y, SNo xSNo yx * (- y) = - (x * y)
Proof:
The rest of this subproof is missing.
End of Section SurrealMul
Beginning of Section Int
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Definition. We define int to be ω {- n|nω} of type set.
Theorem. (int_SNo_cases)
∀p : setprop, (∀nω, p n)(∀nω, p (- n))∀xint, p x
Proof:
The rest of this subproof is missing.
Theorem. (int_3_cases)
∀nint, ∀p : prop, (∀mω, n = - ordsucc mp)(n = 0p)(∀mω, n = ordsucc mp)p
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section Int
Beginning of Section BezoutAndGCD
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Theorem. (quotient_remainder_nat)
∀nω {0}, ∀m, nat_p m∃qω, ∃rn, m = q * n + r
Proof:
The rest of this subproof is missing.
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Theorem. (mul_SNo_nonpos_nonneg)
∀x y, SNo xSNo yx 00 yx * y 0
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (ordinal_ordsucc_pos)
∀alpha, ordinal alpha0 < ordsucc alpha
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define divides_int to be λm n ⇒ m int n int ∃kint, m * k = n of type setsetprop.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (divides_int_mul_SNo)
∀m n m' n', divides_int m m'divides_int n n'divides_int (m * n) (m' * n')
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (divides_int_pos_Le)
∀m n, divides_int m n0 < nm n
Proof:
The rest of this subproof is missing.
Definition. We define gcd_reln to be λm n d ⇒ divides_int d m divides_int d n ∀d', divides_int d' mdivides_int d' nd' d of type setsetsetprop.
Theorem. (gcd_reln_uniq)
∀a b c d, gcd_reln a b cgcd_reln a b dc = d
Proof:
The rest of this subproof is missing.
Definition. We define int_lin_comb to be λa b c ⇒ a int b int c int ∃m nint, m * a + n * b = c of type setsetsetprop.
Theorem. (int_lin_comb_I)
∀a b cint, (∃m nint, m * a + n * b = c)int_lin_comb a b c
Proof:
The rest of this subproof is missing.
Theorem. (int_lin_comb_E)
∀a b c, int_lin_comb a b c∀p : prop, (a intb intc int∀m nint, m * a + n * b = cp)p
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (int_lin_comb_E4)
∀a b c, int_lin_comb a b c∀p : prop, (∀m nint, m * a + n * b = cp)p
Proof:
The rest of this subproof is missing.
Theorem. (least_pos_int_lin_comb_ex)
∀a bint, ¬ (a = 0 b = 0)∃c, int_lin_comb a b c 0 < c ∀c', int_lin_comb a b c'0 < c'c c'
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (least_pos_int_lin_comb_divides_int)
∀a b d, int_lin_comb a b d0 < d(∀c, int_lin_comb a b c0 < cd c)divides_int d a
Proof:
The rest of this subproof is missing.
Theorem. (least_pos_int_lin_comb_gcd)
∀a b d, int_lin_comb a b d0 < d(∀c, int_lin_comb a b c0 < cd c)gcd_reln a b d
Proof:
The rest of this subproof is missing.
Theorem. (BezoutThm)
∀a bint, ¬ (a = 0 b = 0)∀d, gcd_reln a b d int_lin_comb a b d 0 < d ∀d', int_lin_comb a b d'0 < d'd d'
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (gcd_sym)
∀m n d, gcd_reln m n dgcd_reln n m d
Proof:
The rest of this subproof is missing.
Theorem. (gcd_minus)
∀m n d, gcd_reln m n dgcd_reln m (- n) d
Proof:
The rest of this subproof is missing.
Theorem. (euclidean_algorithm_prop_1)
∀m n d, n intgcd_reln m (n + - m) dgcd_reln m n d
Proof:
The rest of this subproof is missing.
Theorem. (euclidean_algorithm)
(∀mω {0}, gcd_reln m m m) (∀mω {0}, gcd_reln 0 m m) (∀mω {0}, gcd_reln m 0 m) (∀m nω, m < n∀d, gcd_reln m (n + - m) dgcd_reln m n d) (∀m nω, n < m∀d, gcd_reln n m dgcd_reln m n d) (∀mω, ∀nint, n < 0∀d, gcd_reln m (- n) dgcd_reln m n d) (∀m nint, m < 0∀d, gcd_reln (- m) n dgcd_reln m n d)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section BezoutAndGCD
Beginning of Section PrimeFactorization
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Proof:
The rest of this subproof is missing.
Definition. We define Pi_SNo to be λf n ⇒ nat_primrec 1 (λi r ⇒ r * f i) n of type (setset)setset.
Theorem. (Pi_SNo_0)
∀f : setset, Pi_SNo f 0 = 1
Proof:
The rest of this subproof is missing.
Theorem. (Pi_SNo_S)
∀f : setset, ∀n, nat_p nPi_SNo f (ordsucc n) = Pi_SNo f n * f n
Proof:
The rest of this subproof is missing.
Theorem. (Pi_SNo_In_omega)
∀f : setset, ∀n, nat_p n(∀in, f i ω)Pi_SNo f n ω
Proof:
The rest of this subproof is missing.
Theorem. (Pi_SNo_In_int)
∀f : setset, ∀n, nat_p n(∀in, f i int)Pi_SNo f n int
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (Euclid_lemma_Pi_SNo)
∀f : setset, ∀p, prime_nat p∀n, nat_p n(∀in, f i int)divides_int p (Pi_SNo f n)∃in, divides_int p (f i)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (Pi_SNo_divides)
∀f : setset, ∀n, nat_p n(∀in, f i ω)(∀in, divides_nat (f i) (Pi_SNo f n))
Proof:
The rest of this subproof is missing.
Definition. We define nonincrfinseq to be λA n f ⇒ ∀in, A (f i) ∀ji, f i f j of type (setprop)set(setset)prop.
Theorem. (Pi_SNo_eq)
∀f g : setset, ∀m, nat_p m(∀im, f i = g i)Pi_SNo f m = Pi_SNo g m
Proof:
The rest of this subproof is missing.
Theorem. (prime_factorization_ex_uniq)
∀n, nat_p n0 n∃kω, ∃f : setset, nonincrfinseq prime_nat k f Pi_SNo f k = n ∀k'ω, ∀f' : setset, nonincrfinseq prime_nat k' f'Pi_SNo f' k' = nk' = k ∀ik, f' i = f i
Proof:
The rest of this subproof is missing.
End of Section PrimeFactorization
Beginning of Section SurrealExp
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Definition. We define exp_SNo_nat to be λn m : setnat_primrec 1 (λ_ r ⇒ n * r) m of type setsetset.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Theorem. (exp_SNo_nat_0)
∀x, SNo xx ^ 0 = 1
Proof:
The rest of this subproof is missing.
Theorem. (exp_SNo_nat_S)
∀x, SNo x∀n, nat_p nx ^ (ordsucc n) = x * x ^ n
Proof:
The rest of this subproof is missing.
Theorem. (exp_SNo_nat_1)
∀x, SNo xx ^ 1 = x
Proof:
The rest of this subproof is missing.
Theorem. (SNo_exp_SNo_nat)
∀x, SNo x∀n, nat_p nSNo (x ^ n)
Proof:
The rest of this subproof is missing.
Theorem. (nat_exp_SNo_nat)
∀x, nat_p x∀n, nat_p nnat_p (x ^ n)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (double_eps_1)
∀x y z, SNo xSNo ySNo zx + x = y + zx = eps_ 1 * (y + z)
Proof:
The rest of this subproof is missing.
Theorem. (exp_SNo_1_bd)
∀x, SNo x1 x∀n, nat_p n1 x ^ n
Proof:
The rest of this subproof is missing.
Theorem. (exp_SNo_2_bd)
∀n, nat_p nn < 2 ^ n
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (exp_SNo_nat_mul_add)
∀x, SNo x∀m, nat_p m∀n, nat_p nx ^ m * x ^ n = x ^ (m + n)
Proof:
The rest of this subproof is missing.
Theorem. (exp_SNo_nat_mul_add')
∀x, SNo x∀m nω, x ^ m * x ^ n = x ^ (m + n)
Proof:
The rest of this subproof is missing.
Theorem. (exp_SNo_nat_pos)
∀x, SNo x0 < x∀n, nat_p n0 < x ^ n
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section SurrealExp
Beginning of Section SNoMaxMin
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Definition. We define SNo_max_of to be λX x ⇒ x X SNo x ∀yX, SNo yy x of type setsetprop.
Definition. We define SNo_min_of to be λX x ⇒ x X SNo x ∀yX, SNo yx y of type setsetprop.
Theorem. (minus_SNo_max_min)
∀X y, (∀xX, SNo x)SNo_max_of X ySNo_min_of {- x|xX} (- y)
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_max_min')
∀X y, (∀xX, SNo x)SNo_max_of {- x|xX} ySNo_min_of X (- y)
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_min_max)
∀X y, (∀xX, SNo x)SNo_min_of X ySNo_max_of {- x|xX} (- y)
Proof:
The rest of this subproof is missing.
Theorem. (double_SNo_max_1)
∀x y, SNo xSNo_max_of (SNoL x) y∀z, SNo zx < zy + z < x + x∃wSNoR z, y + w = x + x
Proof:
The rest of this subproof is missing.
Theorem. (double_SNo_min_1)
∀x y, SNo xSNo_min_of (SNoR x) y∀z, SNo zz < xx + x < y + z∃wSNoL z, y + w = x + x
Proof:
The rest of this subproof is missing.
Theorem. (finite_max_exists)
∀X, (∀xX, SNo x)finite XX 0∃x, SNo_max_of X x
Proof:
The rest of this subproof is missing.
Theorem. (finite_min_exists)
∀X, (∀xX, SNo x)finite XX 0∃x, SNo_min_of X x
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section SNoMaxMin
Beginning of Section DiadicRationals
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Proof:
The rest of this subproof is missing.
Definition. We define diadic_rational_p to be λx ⇒ ∃kω, ∃mint, x = eps_ k * m of type setprop.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section DiadicRationals
Beginning of Section SurrealDiv
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Definition. We define SNoL_pos to be λx ⇒ {wSNoL x|0 < w} of type setset.
Theorem. (SNo_recip_pos_pos)
∀x xi, SNo xSNo xi0 < xx * xi = 10 < xi
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recip_lem1)
∀x x' x'i y y', SNo x0 < xx' SNoL_pos xSNo x'ix' * x'i = 1SNo yx * y < 1SNo y'1 + - x * y' = (1 + - x * y) * (x' + - x) * x'i1 < x * y'
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recip_lem2)
∀x x' x'i y y', SNo x0 < xx' SNoL_pos xSNo x'ix' * x'i = 1SNo y1 < x * ySNo y'1 + - x * y' = (1 + - x * y) * (x' + - x) * x'ix * y' < 1
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recip_lem3)
∀x x' x'i y y', SNo x0 < xx' SNoR xSNo x'ix' * x'i = 1SNo yx * y < 1SNo y'1 + - x * y' = (1 + - x * y) * (x' + - x) * x'ix * y' < 1
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recip_lem4)
∀x x' x'i y y', SNo x0 < xx' SNoR xSNo x'ix' * x'i = 1SNo y1 < x * ySNo y'1 + - x * y' = (1 + - x * y) * (x' + - x) * x'i1 < x * y'
Proof:
The rest of this subproof is missing.
Definition. We define SNo_recipauxset to be λY x X g ⇒ yY{(1 + (x' + - x) * y) * g x'|x'X} of type setsetset(setset)set.
Theorem. (SNo_recipauxset_I)
∀Y x X, ∀g : setset, ∀yY, ∀x'X, (1 + (x' + - x) * y) * g x' SNo_recipauxset Y x X g
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recipauxset_E)
∀Y x X, ∀g : setset, ∀zSNo_recipauxset Y x X g, ∀p : prop, (∀yY, ∀x'X, z = (1 + (x' + - x) * y) * g x'p)p
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recipauxset_ext)
∀Y x X, ∀g h : setset, (∀x'X, g x' = h x')SNo_recipauxset Y x X g = SNo_recipauxset Y x X h
Proof:
The rest of this subproof is missing.
Definition. We define SNo_recipaux to be λx g ⇒ nat_primrec ({0},0) (λk p ⇒ (p 0 SNo_recipauxset (p 0) x (SNoR x) g SNo_recipauxset (p 1) x (SNoL_pos x) g,p 1 SNo_recipauxset (p 0) x (SNoL_pos x) g SNo_recipauxset (p 1) x (SNoR x) g)) of type set(setset)setset.
Theorem. (SNo_recipaux_0)
∀x, ∀g : setset, SNo_recipaux x g 0 = ({0},0)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recipaux_S)
∀x, ∀g : setset, ∀n, nat_p nSNo_recipaux x g (ordsucc n) = (SNo_recipaux x g n 0 SNo_recipauxset (SNo_recipaux x g n 0) x (SNoR x) g SNo_recipauxset (SNo_recipaux x g n 1) x (SNoL_pos x) g,SNo_recipaux x g n 1 SNo_recipauxset (SNo_recipaux x g n 0) x (SNoL_pos x) g SNo_recipauxset (SNo_recipaux x g n 1) x (SNoR x) g)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recipaux_lem1)
∀x, SNo x0 < x∀g : setset, (∀x'SNoS_ (SNoLev x), 0 < x'SNo (g x') x' * g x' = 1)∀k, nat_p k(∀ySNo_recipaux x g k 0, SNo y x * y < 1) (∀ySNo_recipaux x g k 1, SNo y 1 < x * y)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recipaux_lem2)
∀x, SNo x0 < x∀g : setset, (∀x'SNoS_ (SNoLev x), 0 < x'SNo (g x') x' * g x' = 1)SNoCutP (kωSNo_recipaux x g k 0) (kωSNo_recipaux x g k 1)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_recipaux_ext)
∀x, SNo x∀g h : setset, (∀x'SNoS_ (SNoLev x), g x' = h x')∀k, nat_p kSNo_recipaux x g k = SNo_recipaux x h k
Proof:
The rest of this subproof is missing.
Beginning of Section recip_SNo_pos
Let G : set(setset)setλx g ⇒ SNoCut (kωSNo_recipaux x g k 0) (kωSNo_recipaux x g k 1)
Definition. We define recip_SNo_pos to be SNo_rec_i G of type setset.
Proof:
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Proof:
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Proof:
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Proof:
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Proof:
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Proof:
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Proof:
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Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section recip_SNo_pos
Definition. We define recip_SNo to be λx ⇒ if 0 < x then recip_SNo_pos x else if x < 0 then - recip_SNo_pos (- x) else 0 of type setset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
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Proof:
The rest of this subproof is missing.
Theorem. (recip_SNo_invR)
∀x, SNo xx 0x * recip_SNo x = 1
Proof:
The rest of this subproof is missing.
Theorem. (recip_SNo_invL)
∀x, SNo xx 0recip_SNo x * x = 1
Proof:
The rest of this subproof is missing.
Theorem. (mul_SNo_nonzero_cancel_L)
∀x y z, SNo xx 0SNo ySNo zx * y = x * zy = z
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define div_SNo to be λx y ⇒ x * recip_SNo y of type setsetset.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Theorem. (SNo_div_SNo)
∀x y, SNo xSNo ySNo (x :/: y)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (mul_div_SNo_invL)
∀x y, SNo xSNo yy 0(x :/: y) * y = x
Proof:
The rest of this subproof is missing.
Theorem. (mul_div_SNo_invR)
∀x y, SNo xSNo yy 0y * (x :/: y) = x
Proof:
The rest of this subproof is missing.
Theorem. (mul_div_SNo_R)
∀x y z, SNo xSNo ySNo z(x :/: y) * z = (x * z) :/: y
Proof:
The rest of this subproof is missing.
Theorem. (mul_div_SNo_L)
∀x y z, SNo xSNo ySNo zz * (x :/: y) = (z * x) :/: y
Proof:
The rest of this subproof is missing.
Theorem. (div_mul_SNo_invL)
∀x y, SNo xSNo yy 0(x * y) :/: y = x
Proof:
The rest of this subproof is missing.
Theorem. (div_div_SNo)
∀x y z, SNo xSNo ySNo z(x :/: y) :/: z = x :/: (y * z)
Proof:
The rest of this subproof is missing.
Theorem. (mul_div_SNo_both)
∀x y z w, SNo xSNo ySNo zSNo w(x :/: y) * (z :/: w) = (x * z) :/: (y * w)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (div_SNo_pos_pos)
∀x y, SNo xSNo y0 < x0 < y0 < x :/: y
Proof:
The rest of this subproof is missing.
Theorem. (div_SNo_neg_pos)
∀x y, SNo xSNo yx < 00 < yx :/: y < 0
Proof:
The rest of this subproof is missing.
Theorem. (div_SNo_pos_LtL)
∀x y z, SNo xSNo ySNo z0 < yx < z * yx :/: y < z
Proof:
The rest of this subproof is missing.
Theorem. (div_SNo_pos_LtR)
∀x y z, SNo xSNo ySNo z0 < yz * y < xz < x :/: y
Proof:
The rest of this subproof is missing.
Theorem. (div_SNo_pos_LtL')
∀x y z, SNo xSNo ySNo z0 < yx :/: y < zx < z * y
Proof:
The rest of this subproof is missing.
Theorem. (div_SNo_pos_LtR')
∀x y z, SNo xSNo ySNo z0 < yz < x :/: yz * y < x
Proof:
The rest of this subproof is missing.
Theorem. (mul_div_SNo_nonzero_eq)
∀x y z, SNo xSNo ySNo zy 0x = y * zx :/: y = z
Proof:
The rest of this subproof is missing.
End of Section SurrealDiv
Beginning of Section Reals
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define real to be {xSNoS_ (ordsucc ω)|x ω x - ω (∀qSNoS_ ω, (∀kω, abs_SNo (q + - x) < eps_ k)q = x)} of type set.
Theorem. (real_I)
∀xSNoS_ (ordsucc ω), x ωx - ω(∀qSNoS_ ω, (∀kω, abs_SNo (q + - x) < eps_ k)q = x)x real
Proof:
The rest of this subproof is missing.
Theorem. (real_E)
∀xreal, ∀p : prop, (SNo xSNoLev x ordsucc ωx SNoS_ (ordsucc ω)- ω < xx < ω(∀qSNoS_ ω, (∀kω, abs_SNo (q + - x) < eps_ k)q = x)(∀kω, ∃qSNoS_ ω, q < x x < q + eps_ k)p)p
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (real_SNoCut_SNoS_omega)
∀L RSNoS_ ω, SNoCutP L RL 0R 0(∀wL, ∃w'L, w < w')(∀zR, ∃z'R, z' < z)SNoCut L R real
Proof:
The rest of this subproof is missing.
Theorem. (real_SNoCut)
∀L Rreal, SNoCutP L RL 0R 0(∀wL, ∃w'L, w < w')(∀zR, ∃z'R, z' < z)SNoCut L R real
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_prereal_1)
∀x, SNo x(∀qSNoS_ ω, (∀kω, abs_SNo (q + - x) < eps_ k)q = x)(∀qSNoS_ ω, (∀kω, abs_SNo (q + - - x) < eps_ k)q = - x)
Proof:
The rest of this subproof is missing.
Theorem. (minus_SNo_prereal_2)
∀x, SNo x(∀kω, ∃qSNoS_ ω, q < x x < q + eps_ k)(∀kω, ∃qSNoS_ ω, q < - x - x < q + eps_ k)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_prereal_incr_lower_pos)
∀x, SNo x0 < x(∀qSNoS_ ω, (∀kω, abs_SNo (q + - x) < eps_ k)q = x)(∀kω, ∃qSNoS_ ω, q < x x < q + eps_ k)∀kω, ∀p : prop, (∀qSNoS_ ω, 0 < qq < xx < q + eps_ kp)p
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNo_prereal_incr_lower_approx)
∀x, SNo x(∀qSNoS_ ω, (∀kω, abs_SNo (q + - x) < eps_ k)q = x)(∀kω, ∃qSNoS_ ω, q < x x < q + eps_ k)∃fSNoS_ ωω, ∀nω, f n < x x < f n + eps_ n ∀in, f i < f n
Proof:
The rest of this subproof is missing.
Theorem. (SNo_prereal_decr_upper_approx)
∀x, SNo x(∀qSNoS_ ω, (∀kω, abs_SNo (q + - x) < eps_ k)q = x)(∀kω, ∃qSNoS_ ω, q < x x < q + eps_ k)∃gSNoS_ ωω, ∀nω, g n + - eps_ n < x x < g n ∀in, g n < g i
Proof:
The rest of this subproof is missing.
Theorem. (SNoCutP_SNoCut_lim)
∀lambda, ordinal lambda(∀alphalambda, ordsucc alpha lambda)∀L RSNoS_ lambda, SNoCutP L RSNoLev (SNoCut L R) ordsucc lambda
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (SNo_approx_real_lem)
∀f gSNoS_ ωω, (∀n mω, f n < g m)∀p : prop, (SNoCutP {f n|nω} {g n|nω}SNo (SNoCut {f n|nω} {g n|nω})SNoLev (SNoCut {f n|nω} {g n|nω}) ordsucc ωSNoCut {f n|nω} {g n|nω} SNoS_ (ordsucc ω)(∀nω, f n < SNoCut {f n|nω} {g n|nω})(∀nω, SNoCut {f n|nω} {g n|nω} < g n)p)p
Proof:
The rest of this subproof is missing.
Theorem. (SNo_approx_real)
∀x, SNo x∀f gSNoS_ ωω, (∀nω, f n < x)(∀nω, x < f n + eps_ n)(∀nω, ∀in, f i < f n)(∀nω, x < g n)(∀nω, ∀in, g n < g i)x = SNoCut {f n|nω} {g n|nω}x real
Proof:
The rest of this subproof is missing.
Theorem. (SNo_approx_real_rep)
∀xreal, ∀p : prop, (∀f gSNoS_ ωω, (∀nω, f n < x)(∀nω, x < f n + eps_ n)(∀nω, ∀in, f i < f n)(∀nω, g n + - eps_ n < x)(∀nω, x < g n)(∀nω, ∀in, g n < g i)SNoCutP {f n|nω} {g n|nω}x = SNoCut {f n|nω} {g n|nω}p)p
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (real_mul_SNo_pos)
∀x yreal, 0 < x0 < yx * y real
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (pos_real_left_approx_double)
∀xreal, 0 < xx 2(∀mω, x eps_ m)∃wSNoL_pos x, x < 2 * w
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section Reals
Beginning of Section even_odd
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Theorem. (nat_le2_cases)
∀m, nat_p mm 2m = 0 m = 1 m = 2
Proof:
The rest of this subproof is missing.
Theorem. (prime_nat_2_lem)
∀m, nat_p m∀n, nat_p nm * n = 2m = 1 m = 2
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Proof:
The rest of this subproof is missing.
End of Section even_odd
Beginning of Section form100_22b
Let tag : setsetλalpha ⇒ SetAdjoin alpha {1}
Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag.
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Proof:
The rest of this subproof is missing.
Theorem. (Repl_finite)
∀f : setset, ∀X, finite Xfinite {f x|xX}
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section form100_22b
Beginning of Section rational
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Definition. We define rational to be {xreal|∃mint, ∃nω {0}, x = m :/: n} of type set.
End of Section rational
Beginning of Section form100_3
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define nat_pair to be λm n ⇒ 2 ^ m * (2 * n + 1) of type setsetset.
Proof:
The rest of this subproof is missing.
Theorem. (nat_pair_0)
∀m n m' n'ω, nat_pair m n = nat_pair m' n'm = m'
Proof:
The rest of this subproof is missing.
Theorem. (nat_pair_1)
∀m n m' n'ω, nat_pair m n = nat_pair m' n'n = n'
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section form100_3
Beginning of Section SurrealSqrt
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Definition. We define SNoL_nonneg to be λx ⇒ {wSNoL x|0 w} of type setset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define SNo_sqrtauxset to be λY Z x ⇒ yY{(x + y * z) :/: (y + z)|zZ, 0 < y + z} of type setsetsetset.
Theorem. (SNo_sqrtauxset_I)
∀Y Z x, ∀yY, ∀zZ, 0 < y + z(x + y * z) :/: (y + z) SNo_sqrtauxset Y Z x
Proof:
The rest of this subproof is missing.
Theorem. (SNo_sqrtauxset_E)
∀Y Z x, ∀uSNo_sqrtauxset Y Z x, ∀p : prop, (∀yY, ∀zZ, 0 < y + zu = (x + y * z) :/: (y + z)p)p
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define SNo_sqrtaux to be λx g ⇒ nat_primrec ({g w|wSNoL_nonneg x},{g z|zSNoR x}) (λk p ⇒ (p 0 SNo_sqrtauxset (p 0) (p 1) x,p 1 SNo_sqrtauxset (p 0) (p 0) x SNo_sqrtauxset (p 1) (p 1) x)) of type set(setset)setset.
Theorem. (SNo_sqrtaux_0)
∀x, ∀g : setset, SNo_sqrtaux x g 0 = ({g w|wSNoL_nonneg x},{g z|zSNoR x})
Proof:
The rest of this subproof is missing.
Theorem. (SNo_sqrtaux_S)
∀x, ∀g : setset, ∀n, nat_p nSNo_sqrtaux x g (ordsucc n) = (SNo_sqrtaux x g n 0 SNo_sqrtauxset (SNo_sqrtaux x g n 0) (SNo_sqrtaux x g n 1) x,SNo_sqrtaux x g n 1 SNo_sqrtauxset (SNo_sqrtaux x g n 0) (SNo_sqrtaux x g n 0) x SNo_sqrtauxset (SNo_sqrtaux x g n 1) (SNo_sqrtaux x g n 1) x)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_sqrtaux_mon_lem)
∀x, ∀g : setset, ∀m, nat_p m∀n, nat_p nSNo_sqrtaux x g m 0 SNo_sqrtaux x g (add_nat m n) 0 SNo_sqrtaux x g m 1 SNo_sqrtaux x g (add_nat m n) 1
Proof:
The rest of this subproof is missing.
Theorem. (SNo_sqrtaux_mon)
∀x, ∀g : setset, ∀m, nat_p m∀n, nat_p nm nSNo_sqrtaux x g m 0 SNo_sqrtaux x g n 0 SNo_sqrtaux x g m 1 SNo_sqrtaux x g n 1
Proof:
The rest of this subproof is missing.
Theorem. (SNo_sqrtaux_ext)
∀x, SNo x∀g h : setset, (∀x'SNoS_ (SNoLev x), g x' = h x')∀k, nat_p kSNo_sqrtaux x g k = SNo_sqrtaux x h k
Proof:
The rest of this subproof is missing.
Beginning of Section sqrt_SNo_nonneg
Let G : set(setset)setλx g ⇒ SNoCut (kωSNo_sqrtaux x g k 0) (kωSNo_sqrtaux x g k 1)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section sqrt_SNo_nonneg
Theorem. (SNo_sqrtaux_0_1_prop)
∀x, SNo x0 x∀k, nat_p k(∀ySNo_sqrtaux x sqrt_SNo_nonneg k 0, SNo y 0 y y * y < x) (∀ySNo_sqrtaux x sqrt_SNo_nonneg k 1, SNo y 0 y x < y * y)
Proof:
The rest of this subproof is missing.
Theorem. (SNo_sqrtaux_0_prop)
∀x, SNo x0 x∀k, nat_p k∀ySNo_sqrtaux x sqrt_SNo_nonneg k 0, SNo y 0 y y * y < x
Proof:
The rest of this subproof is missing.
Theorem. (SNo_sqrtaux_1_prop)
∀x, SNo x0 x∀k, nat_p k∀ySNo_sqrtaux x sqrt_SNo_nonneg k 1, SNo y 0 y x < y * y
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
End of Section SurrealSqrt
Beginning of Section form100_1
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Proof:
The rest of this subproof is missing.
Theorem. (form100_1_lem1)
∀m, nat_p m∀n, nat_p nm * m = 2 * n * nn = 0
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Proof:
The rest of this subproof is missing.
End of Section form100_1
Beginning of Section Topology
Definition. We define topology_on to be λX T ⇒ T 𝒫 X Empty T X T (∀UFam𝒫 T, UFam T) (∀UT, ∀VT, U V T) of type setsetprop.
Definition. We define open_in to be λX T U ⇒ topology_on X T U T of type setsetsetprop.
Definition. We define closed_in to be λX T C ⇒ topology_on X T ∃UT, C = X U of type setsetsetprop.
Theorem. (closed_of_open_complement)
∀X T U : set, topology_on X TU Tclosed_in X T (X U)
Proof:
Let X, T and U be given.
Assume Htop: topology_on X T.
Assume HU: U T.
We will prove topology_on X T ∃U0T, X U = X U0.
Apply andI to the current goal.
An exact proof term for the current goal is Htop.
We use U to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HU.
Use reflexivity.
Definition. We define finer_than to be λT' T ⇒ T T' of type setsetprop.
Definition. We define coarser_than to be λT' T ⇒ T' T of type setsetprop.
Definition. We define discrete_topology to be λX ⇒ 𝒫 X of type setset.
Definition. We define indiscrete_topology to be λX ⇒ {Empty,X} of type setset.
Definition. We define finite_complement_topology to be λX ⇒ {U𝒫 X|finite (X U) U = Empty} of type setset.
Definition. We define countable to be λX ⇒ atleastp X ω of type setprop.
Proof:
Let X be given.
Assume Hfin.
Apply Hfin to the current goal.
Let n be given.
Assume Hpair: n ω equip X n.
We prove the intermediate claim Hn: n ω.
An exact proof term for the current goal is (andEL (n ω) (equip X n) Hpair).
We prove the intermediate claim Heq: equip X n.
An exact proof term for the current goal is (andER (n ω) (equip X n) Hpair).
We prove the intermediate claim Hn_sub: n ω.
An exact proof term for the current goal is (omega_TransSet n Hn).
We prove the intermediate claim Hcount_n: atleastp n ω.
An exact proof term for the current goal is (Subq_atleastp n ω Hn_sub).
We prove the intermediate claim Hcount_X: atleastp X n.
An exact proof term for the current goal is (equip_atleastp X n Heq).
An exact proof term for the current goal is (atleastp_tra X n ω Hcount_X Hcount_n).
Definition. We define countable_complement_topology to be λX ⇒ {U𝒫 X|countable (X U) U = Empty} of type setset.
Proof:
Let X be given.
We will prove 𝒫 X 𝒫 X Empty 𝒫 X X 𝒫 X (∀UFam𝒫 (𝒫 X), UFam 𝒫 X) (∀U𝒫 X, ∀V𝒫 X, U V 𝒫 X).
Apply andI to the current goal.
We will prove ((𝒫 X 𝒫 X Empty 𝒫 X) X 𝒫 X (∀UFam𝒫 (𝒫 X), UFam 𝒫 X)).
Apply andI to the current goal.
We will prove 𝒫 X 𝒫 X Empty 𝒫 X X 𝒫 X.
Apply andI to the current goal.
We will prove 𝒫 X 𝒫 X Empty 𝒫 X.
Apply andI to the current goal.
An exact proof term for the current goal is (Subq_ref (𝒫 X)).
Apply Empty_In_Power to the current goal.
Apply PowerI to the current goal.
An exact proof term for the current goal is (Subq_ref X).
We will prove ∀UFam𝒫 (𝒫 X), UFam 𝒫 X.
Let UFam be given.
Assume Hfam: UFam 𝒫 (𝒫 X).
Apply PowerI X ( UFam) to the current goal.
Let x be given.
Assume HxUnion: x UFam.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HUinx: x U.
Assume HUinFam: U UFam.
We prove the intermediate claim HFamSub: UFam 𝒫 X.
An exact proof term for the current goal is (iffEL (UFam 𝒫 (𝒫 X)) (UFam 𝒫 X) (PowerEq (𝒫 X) UFam) Hfam).
We prove the intermediate claim HUinPower: U 𝒫 X.
An exact proof term for the current goal is HFamSub U HUinFam.
We prove the intermediate claim HUsub: U X.
An exact proof term for the current goal is (iffEL (U 𝒫 X) (U X) (PowerEq X U) HUinPower).
An exact proof term for the current goal is (HUsub x HUinx).
We will prove ∀U𝒫 X, ∀V𝒫 X, U V 𝒫 X.
Let U be given.
Assume HU: U 𝒫 X.
Let V be given.
Assume HV: V 𝒫 X.
Apply PowerI X (U V) to the current goal.
Let x be given.
Assume Hxcap: x U V.
Apply binintersectE U V x Hxcap to the current goal.
Assume HxU HxV.
We prove the intermediate claim HUsub: U X.
An exact proof term for the current goal is (iffEL (U 𝒫 X) (U X) (PowerEq X U) HU).
An exact proof term for the current goal is (HUsub x HxU).
Proof:
Let X be given.
Apply andI to the current goal.
Apply andI to the current goal.
Apply andI to the current goal.
Apply andI to the current goal.
Let U be given.
Assume HU: U indiscrete_topology X.
Apply UPairE U Empty X HU to the current goal.
Assume HUe: U = Empty.
rewrite the current goal using HUe (from left to right).
An exact proof term for the current goal is (Empty_In_Power X).
Assume HUX: U = X.
rewrite the current goal using HUX (from left to right).
An exact proof term for the current goal is (Self_In_Power X).
An exact proof term for the current goal is (UPairI1 Empty X).
An exact proof term for the current goal is (UPairI2 Empty X).
We will prove ∀UFam𝒫 (indiscrete_topology X), UFam indiscrete_topology X.
Let UFam be given.
Assume Hfam: UFam 𝒫 (indiscrete_topology X).
We prove the intermediate claim Hsub: UFam indiscrete_topology X.
An exact proof term for the current goal is (PowerE (indiscrete_topology X) UFam Hfam).
Apply xm (∃U : set, U UFam U = X) to the current goal.
Assume Hex: ∃U : set, U UFam U = X.
We prove the intermediate claim HUnion_sub: UFam X.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUtop: U indiscrete_topology X.
An exact proof term for the current goal is (Hsub U HUin).
Apply UPairE U Empty X HUtop to the current goal.
Assume HUe: U = Empty.
We prove the intermediate claim HxEmpty: x Empty.
rewrite the current goal using HUe (from right to left).
An exact proof term for the current goal is HxU.
An exact proof term for the current goal is (EmptyE x HxEmpty (x X)).
Assume HUX: U = X.
rewrite the current goal using HUX (from right to left).
An exact proof term for the current goal is HxU.
We prove the intermediate claim HX_sub: X UFam.
Let x be given.
Assume HxX.
Apply Hex to the current goal.
Let U be given.
Assume HUinpair: U UFam U = X.
We prove the intermediate claim HUin: U UFam.
An exact proof term for the current goal is (andEL (U UFam) (U = X) HUinpair).
We prove the intermediate claim HUeq: U = X.
An exact proof term for the current goal is (andER (U UFam) (U = X) HUinpair).
We prove the intermediate claim HxU: x U.
rewrite the current goal using HUeq (from left to right).
An exact proof term for the current goal is HxX.
Apply UnionI UFam x U HxU HUin to the current goal.
We prove the intermediate claim HUnion_eq: UFam = X.
Apply set_ext to the current goal.
An exact proof term for the current goal is HUnion_sub.
An exact proof term for the current goal is HX_sub.
rewrite the current goal using HUnion_eq (from left to right).
An exact proof term for the current goal is (UPairI2 Empty X).
Assume Hnone: ¬ ∃U : set, U UFam U = X.
We prove the intermediate claim HUnion_empty: UFam = Empty.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUtop: U indiscrete_topology X.
An exact proof term for the current goal is (Hsub U HUin).
Apply UPairE U Empty X HUtop to the current goal.
Assume HUe: U = Empty.
We prove the intermediate claim HxEmpty: x Empty.
rewrite the current goal using HUe (from right to left).
An exact proof term for the current goal is HxU.
An exact proof term for the current goal is HxEmpty.
Assume HUX: U = X.
Apply FalseE to the current goal.
Apply Hnone to the current goal.
We use U to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HUin.
An exact proof term for the current goal is HUX.
rewrite the current goal using HUnion_empty (from left to right).
An exact proof term for the current goal is (UPairI1 Empty X).
Let U be given.
Assume HU: U indiscrete_topology X.
Let V be given.
Assume HV: V indiscrete_topology X.
Apply UPairE U Empty X HU to the current goal.
Assume HUe: U = Empty.
We prove the intermediate claim Hcap: U V = Empty.
rewrite the current goal using HUe (from left to right).
Apply Empty_Subq_eq to the current goal.
An exact proof term for the current goal is (binintersect_Subq_1 Empty V).
rewrite the current goal using Hcap (from left to right).
An exact proof term for the current goal is (UPairI1 Empty X).
Assume HUX: U = X.
Apply UPairE V Empty X HV to the current goal.
Assume HVe: V = Empty.
We prove the intermediate claim Hcap: U V = Empty.
rewrite the current goal using HVe (from left to right).
Apply Empty_Subq_eq to the current goal.
An exact proof term for the current goal is (binintersect_Subq_2 U Empty).
rewrite the current goal using Hcap (from left to right).
An exact proof term for the current goal is (UPairI1 Empty X).
Assume HVX: V = X.
We prove the intermediate claim Hcap: U V = X.
Apply set_ext to the current goal.
rewrite the current goal using HUX (from left to right).
rewrite the current goal using HVX (from left to right).
An exact proof term for the current goal is (binintersect_Subq_1 X X).
Let x be given.
Assume HxX.
rewrite the current goal using HUX (from left to right).
rewrite the current goal using HVX (from left to right).
An exact proof term for the current goal is (binintersectI X X x HxX HxX).
rewrite the current goal using Hcap (from left to right).
An exact proof term for the current goal is (UPairI2 Empty X).
Proof:
Let X be given.
We prove the intermediate claim HEmptyOpen: Empty finite_complement_topology X.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) Empty (Empty_In_Power X) (orIR (finite (X Empty)) (Empty = Empty) (λP H ⇒ H))).
Apply andI to the current goal.
Apply andI to the current goal.
Apply andI to the current goal.
Apply andI to the current goal.
Let U be given.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U HU).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) Empty (Empty_In_Power X) (orIR (finite (X Empty)) (Empty = Empty) (λP H ⇒ H))).
We prove the intermediate claim Hdiff_empty: X X = Empty.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume Hx.
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (setminusE1 X X x Hx).
We prove the intermediate claim Hxnot: x X.
An exact proof term for the current goal is (setminusE2 X X x Hx).
Apply FalseE to the current goal.
An exact proof term for the current goal is (Hxnot HxX).
We prove the intermediate claim HfinDiff: finite (X X).
rewrite the current goal using Hdiff_empty (from left to right).
An exact proof term for the current goal is finite_Empty.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) X (Self_In_Power X) (orIL (finite (X X)) (X = Empty) HfinDiff)).
Let UFam be given.
Assume Hfam: UFam 𝒫 (finite_complement_topology X).
We prove the intermediate claim Hsub: UFam finite_complement_topology X.
An exact proof term for the current goal is (PowerE (finite_complement_topology X) UFam Hfam).
Apply xm (∃U : set, U UFam finite (X U)) to the current goal.
Assume Hex: ∃U : set, U UFam finite (X U).
We prove the intermediate claim HUnionInPower: UFam 𝒫 X.
Apply PowerI X ( UFam) to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUinPow: U 𝒫 X.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U (Hsub U HUin)).
We prove the intermediate claim HUsub: U X.
An exact proof term for the current goal is (PowerE X U HUinPow).
An exact proof term for the current goal is (HUsub x HxU).
We prove the intermediate claim HUnionPred: finite (X UFam) UFam = Empty.
Apply orIL to the current goal.
Apply Hex to the current goal.
Let U be given.
Assume Hpair: U UFam finite (X U).
We prove the intermediate claim HUin: U UFam.
An exact proof term for the current goal is (andEL (U UFam) (finite (X U)) Hpair).
We prove the intermediate claim HUfin: finite (X U).
An exact proof term for the current goal is (andER (U UFam) (finite (X U)) Hpair).
We prove the intermediate claim Hsubset: X UFam X U.
Let x be given.
Assume Hx.
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (setminusE1 X ( UFam) x Hx).
We prove the intermediate claim HnotUnion: x UFam.
An exact proof term for the current goal is (setminusE2 X ( UFam) x Hx).
We prove the intermediate claim HnotU: x U.
Assume HxU.
Apply HnotUnion to the current goal.
Apply UnionI UFam x U HxU HUin to the current goal.
Apply setminusI X U x HxX HnotU to the current goal.
An exact proof term for the current goal is (Subq_finite (X U) HUfin (X UFam) Hsubset).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) ( UFam) HUnionInPower HUnionPred).
Assume Hnone: ¬ ∃U : set, U UFam finite (X U).
We prove the intermediate claim HUnionEmpty: UFam = Empty.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUdata: finite (X U) U = Empty.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U (Hsub U HUin)).
Apply HUdata (x Empty) to the current goal.
Assume HUfin.
Apply FalseE to the current goal.
Apply Hnone to the current goal.
We use U to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HUin.
An exact proof term for the current goal is HUfin.
Assume HUempty: U = Empty.
rewrite the current goal using HUempty (from right to left).
An exact proof term for the current goal is HxU.
rewrite the current goal using HUnionEmpty (from left to right).
An exact proof term for the current goal is HEmptyOpen.
Let U be given.
Let V be given.
We prove the intermediate claim HUdata: finite (X U) U = Empty.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U HU).
We prove the intermediate claim HVdata: finite (X V) V = Empty.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) V HV).
Apply HUdata (U V finite_complement_topology X) to the current goal.
Assume HUfin.
Apply HVdata (U V finite_complement_topology X) to the current goal.
Assume HVfin.
We prove the intermediate claim HcapInPower: U V 𝒫 X.
We prove the intermediate claim HUsub: U X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U HU)).
Apply PowerI X (U V) to the current goal.
Let x be given.
Assume HxCap.
Apply binintersectE U V x HxCap to the current goal.
Assume HxU HxV.
An exact proof term for the current goal is (HUsub x HxU).
We prove the intermediate claim HcapPred: finite (X (U V)) U V = Empty.
Apply orIL to the current goal.
We prove the intermediate claim HfinUnion: finite ((X U) (X V)).
An exact proof term for the current goal is (binunion_finite (X U) HUfin (X V) HVfin).
We prove the intermediate claim Hsubset: X (U V) (X U) (X V).
Let x be given.
Assume Hx.
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (setminusE1 X (U V) x Hx).
We prove the intermediate claim HnotCap: x U V.
An exact proof term for the current goal is (setminusE2 X (U V) x Hx).
Apply xm (x U) to the current goal.
Assume HxU.
We prove the intermediate claim HnotV: x V.
Assume HxV.
Apply HnotCap to the current goal.
An exact proof term for the current goal is (binintersectI U V x HxU HxV).
Apply binunionI2 (X U) (X V) to the current goal.
Apply setminusI X V x HxX HnotV to the current goal.
Assume HnotU.
Apply binunionI1 (X U) (X V) to the current goal.
Apply setminusI X U x HxX HnotU to the current goal.
An exact proof term for the current goal is (Subq_finite ((X U) (X V)) HfinUnion (X (U V)) Hsubset).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) (U V) HcapInPower HcapPred).
Assume HVempty: V = Empty.
We prove the intermediate claim Hcap_empty: U V = Empty.
rewrite the current goal using HVempty (from left to right).
Apply Empty_Subq_eq to the current goal.
An exact proof term for the current goal is (binintersect_Subq_2 U Empty).
rewrite the current goal using Hcap_empty (from left to right).
An exact proof term for the current goal is HEmptyOpen.
Assume HUempty: U = Empty.
We prove the intermediate claim Hcap_empty: U V = Empty.
rewrite the current goal using HUempty (from left to right).
Apply Empty_Subq_eq to the current goal.
An exact proof term for the current goal is (binintersect_Subq_1 Empty V).
rewrite the current goal using Hcap_empty (from left to right).
An exact proof term for the current goal is HEmptyOpen.
Proof:
Let T be given.
An exact proof term for the current goal is (Subq_ref T).
Theorem. (finer_than_trans)
∀A B C : set, finer_than B Afiner_than C Bfiner_than C A
Proof:
Let A, B and C be given.
Assume H1: finer_than B A.
Assume H2: finer_than C B.
An exact proof term for the current goal is (Subq_tra A B C H1 H2).
Proof:
Let T and T' be given.
Assume H.
An exact proof term for the current goal is H.
Definition. We define comparable_topologies to be λT1 T2 ⇒ finer_than T1 T2 finer_than T2 T1 of type setsetprop.
Definition. We define topology_eq to be λX T1 T2 ⇒ topology_on X T1 topology_on X T2 T1 = T2 of type setsetsetprop.
Theorem. (topology_eq_sym)
∀X T1 T2 : set, topology_eq X T1 T2topology_eq X T2 T1
Proof:
Let X, T1 and T2 be given.
Assume H.
We prove the intermediate claim Hpair: topology_on X T1 topology_on X T2.
An exact proof term for the current goal is (andEL (topology_on X T1 topology_on X T2) (T1 = T2) H).
We prove the intermediate claim Heq: T1 = T2.
An exact proof term for the current goal is (andER (topology_on X T1 topology_on X T2) (T1 = T2) H).
We prove the intermediate claim HT1: topology_on X T1.
An exact proof term for the current goal is (andEL (topology_on X T1) (topology_on X T2) Hpair).
We prove the intermediate claim HT2: topology_on X T2.
An exact proof term for the current goal is (andER (topology_on X T1) (topology_on X T2) Hpair).
We will prove topology_on X T2 topology_on X T1 T2 = T1.
Apply andI to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is HT2.
An exact proof term for the current goal is HT1.
rewrite the current goal using Heq (from right to left).
Use reflexivity.
Theorem. (topology_eq_trans)
∀X T1 T2 T3 : set, topology_eq X T1 T2topology_eq X T2 T3topology_eq X T1 T3
Proof:
Let X, T1, T2 and T3 be given.
Assume H12 H23.
We prove the intermediate claim H12pair: topology_on X T1 topology_on X T2.
An exact proof term for the current goal is (andEL (topology_on X T1 topology_on X T2) (T1 = T2) H12).
We prove the intermediate claim H12eq: T1 = T2.
An exact proof term for the current goal is (andER (topology_on X T1 topology_on X T2) (T1 = T2) H12).
We prove the intermediate claim HT1: topology_on X T1.
An exact proof term for the current goal is (andEL (topology_on X T1) (topology_on X T2) H12pair).
We prove the intermediate claim HT2: topology_on X T2.
An exact proof term for the current goal is (andER (topology_on X T1) (topology_on X T2) H12pair).
We prove the intermediate claim H23pair: topology_on X T2 topology_on X T3.
An exact proof term for the current goal is (andEL (topology_on X T2 topology_on X T3) (T2 = T3) H23).
We prove the intermediate claim H23eq: T2 = T3.
An exact proof term for the current goal is (andER (topology_on X T2 topology_on X T3) (T2 = T3) H23).
We prove the intermediate claim HT3: topology_on X T3.
An exact proof term for the current goal is (andER (topology_on X T2) (topology_on X T3) H23pair).
We will prove topology_on X T1 topology_on X T3 T1 = T3.
Apply andI to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is HT1.
An exact proof term for the current goal is HT3.
rewrite the current goal using H12eq (from left to right).
rewrite the current goal using H23eq (from left to right).
Use reflexivity.
Proof:
Let X and T be given.
Assume HT.
We will prove topology_on X T topology_on X T T = T.
Apply andI to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is HT.
An exact proof term for the current goal is HT.
Use reflexivity.
Definition. We define strictly_finer_than to be λT' T ⇒ finer_than T' T ¬ finer_than T T' of type setsetprop.
Definition. We define strictly_coarser_than to be λT' T ⇒ coarser_than T' T ¬ coarser_than T T' of type setsetprop.
Definition. We define discrete_topology_alt to be discrete_topology of type setset.
Definition. We define trivial_topology to be indiscrete_topology of type setset.
Definition. We define finer_than_topology to be λX T' T ⇒ topology_on X T' topology_on X T finer_than T' T of type setsetsetprop.
Proof:
Let T and T' be given.
Apply iffI to the current goal.
Assume H.
An exact proof term for the current goal is H.
Assume H.
An exact proof term for the current goal is H.
Proof:
Let X and T be given.
Assume HT.
We prove the intermediate claim H1: ((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T).
An exact proof term for the current goal is (andEL (((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T)) (∀UT, ∀VT, U V T) HT).
We prove the intermediate claim H2: (T 𝒫 X Empty T) X T.
An exact proof term for the current goal is (andEL ((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T) H1).
We prove the intermediate claim H3: T 𝒫 X Empty T.
An exact proof term for the current goal is (andEL (T 𝒫 X Empty T) (X T) H2).
We prove the intermediate claim HTsub: T 𝒫 X.
An exact proof term for the current goal is (andEL (T 𝒫 X) (Empty T) H3).
An exact proof term for the current goal is HTsub.
Proof:
Let X and T be given.
Assume HT.
We prove the intermediate claim Hchunk1: ((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T).
An exact proof term for the current goal is (andEL (((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T)) (∀UT, ∀VT, U V T) HT).
We prove the intermediate claim Hchunk2: (T 𝒫 X Empty T) X T.
An exact proof term for the current goal is (andEL ((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T) Hchunk1).
We prove the intermediate claim Hchunk3: T 𝒫 X Empty T.
An exact proof term for the current goal is (andEL (T 𝒫 X Empty T) (X T) Hchunk2).
We prove the intermediate claim Hempty: Empty T.
An exact proof term for the current goal is (andER (T 𝒫 X) (Empty T) Hchunk3).
We prove the intermediate claim HX: X T.
An exact proof term for the current goal is (andER ((T 𝒫 X) Empty T) (X T) Hchunk2).
Let U be given.
Assume HU: U indiscrete_topology X.
Apply UPairE U Empty X HU to the current goal.
Assume HUempty: U = Empty.
rewrite the current goal using HUempty (from left to right).
An exact proof term for the current goal is Hempty.
Assume HUX: U = X.
rewrite the current goal using HUX (from left to right).
An exact proof term for the current goal is HX.
Proof:
Let X and U be given.
Assume HUsub.
Apply PowerI X U HUsub to the current goal.
Proof:
Let X and U be given.
Apply iffI to the current goal.
Assume HU.
An exact proof term for the current goal is (UPairE U Empty X HU).
Assume Hcases: U = Empty U = X.
We prove the intermediate claim HUempty_branch: U = EmptyU indiscrete_topology X.
Assume HUE: U = Empty.
rewrite the current goal using HUE (from left to right).
An exact proof term for the current goal is (UPairI1 Empty X).
We prove the intermediate claim HUx_branch: U = XU indiscrete_topology X.
Assume HUX: U = X.
rewrite the current goal using HUX (from left to right).
An exact proof term for the current goal is (UPairI2 Empty X).
An exact proof term for the current goal is (Hcases (U indiscrete_topology X) HUempty_branch HUx_branch).
Proof:
Let X and U be given.
Assume Hopen.
We prove the intermediate claim HUin: U finite_complement_topology X.
An exact proof term for the current goal is (andER (topology_on X (finite_complement_topology X)) (U finite_complement_topology X) Hopen).
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U HUin).
Proof:
Let X be given.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) Empty (Empty_In_Power X) (orIR (finite (X Empty)) (Empty = Empty) (λP H ⇒ H))).
Proof:
Let X be given.
We prove the intermediate claim Hdiff_empty: X X = Empty.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume Hx.
We prove the intermediate claim Hxin: x X.
An exact proof term for the current goal is (setminusE1 X X x Hx).
We prove the intermediate claim Hxnot: x X.
An exact proof term for the current goal is (setminusE2 X X x Hx).
Apply FalseE to the current goal.
An exact proof term for the current goal is (Hxnot Hxin).
We prove the intermediate claim HfinDiff: finite (X X).
rewrite the current goal using Hdiff_empty (from left to right).
An exact proof term for the current goal is finite_Empty.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) X (Self_In_Power X) (orIL (finite (X X)) (X = Empty) HfinDiff)).
Proof:
Let X and U be given.
Assume Hopen.
We prove the intermediate claim HUin: U countable_complement_topology X.
An exact proof term for the current goal is (andER (topology_on X (countable_complement_topology X)) (U countable_complement_topology X) Hopen).
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : setcountable (X U0) U0 = Empty) U HUin).
Proof:
Let X be given.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setcountable (X U0) U0 = Empty) Empty (Empty_In_Power X) (orIR (countable (X Empty)) (Empty = Empty) (λP H ⇒ H))).
Proof:
Let X be given.
We prove the intermediate claim Hdiff_empty: X X = Empty.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume Hx.
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (setminusE1 X X x Hx).
We prove the intermediate claim Hxnot: x X.
An exact proof term for the current goal is (setminusE2 X X x Hx).
Apply FalseE to the current goal.
An exact proof term for the current goal is (Hxnot HxX).
We prove the intermediate claim HcountDiff: countable (X X).
rewrite the current goal using Hdiff_empty (from left to right).
An exact proof term for the current goal is (Subq_atleastp Empty ω (Subq_Empty ω)).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setcountable (X U0) U0 = Empty) X (Self_In_Power X) (orIL (countable (X X)) (X = Empty) HcountDiff)).
Proof:
Let X be given.
Let U be given.
We prove the intermediate claim HUinPow: U 𝒫 X.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U HU).
We prove the intermediate claim HUdata: finite (X U) U = Empty.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U HU).
We prove the intermediate claim HUpred: countable (X U) U = Empty.
Apply HUdata (countable (X U) U = Empty) to the current goal.
Assume HUfin: finite (X U).
Apply orIL to the current goal.
An exact proof term for the current goal is (finite_countable (X U) HUfin).
Assume HUemp: U = Empty.
Apply orIR to the current goal.
An exact proof term for the current goal is HUemp.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : setcountable (X U0) U0 = Empty) U HUinPow HUpred).
Proof:
Let X be given.
Let U be given.
Assume HU.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : setfinite (X U0) U0 = Empty) U HU).
Proof:
Let X be given.
Let U be given.
Assume HU: U indiscrete_topology X.
Apply UPairE U Empty X HU to the current goal.
Assume HUempty: U = Empty.
rewrite the current goal using HUempty (from left to right).
An exact proof term for the current goal is (countable_complement_topology_contains_empty X).
Assume HUX: U = X.
rewrite the current goal using HUX (from left to right).
An exact proof term for the current goal is (countable_complement_topology_contains_full X).
Definition. We define finer_than_topology_by_inclusion to be λX T' T ⇒ topology_on X T' topology_on X T T T' of type setsetsetprop.
Proof:
Let X, T' and T be given.
Apply iffI to the current goal.
Assume H.
An exact proof term for the current goal is H.
Assume H.
An exact proof term for the current goal is H.
Theorem. (lemma_union_of_topology_family_open)
∀X T UFam : set, topology_on X TUFam 𝒫 T UFam T
Proof:
Let X, T and UFam be given.
Assume HT Hfam.
We prove the intermediate claim Hchunk1: ((T 𝒫 X Empty T) X T) (∀UFam'𝒫 T, UFam' T).
An exact proof term for the current goal is (andEL (((T 𝒫 X Empty T) X T) (∀UFam'𝒫 T, UFam' T)) (∀UT, ∀VT, U V T) HT).
We prove the intermediate claim Hunion_axiom: ∀UFam'𝒫 T, UFam' T.
An exact proof term for the current goal is (andER ((T 𝒫 X Empty T) X T) (∀UFam'𝒫 T, UFam' T) Hchunk1).
An exact proof term for the current goal is (Hunion_axiom UFam Hfam).
Theorem. (lemma_intersection_two_open)
∀X T U V : set, topology_on X TU TV TU V T
Proof:
Let X, T, U and V be given.
Assume HT HU HV.
We prove the intermediate claim Hax_inter: ∀U'T, ∀V'T, U' V' T.
An exact proof term for the current goal is (andER ((T 𝒫 X Empty T X T (∀UFam𝒫 T, UFam T))) (∀U'T, ∀V'T, U' V' T) HT).
An exact proof term for the current goal is (Hax_inter U HU V HV).
Definition. We define topological_space to be topology_on of type setsetprop.
Definition. We define open_set_family to be λ_ T ⇒ T of type setsetset.
Definition. We define open_set to be λX T U ⇒ topology_on X T U T of type setsetsetprop.
Definition. We define basis_on to be λX B ⇒ B 𝒫 X (∀xX, ∃bB, x b) (∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2) of type setsetprop.
Definition. We define generated_topology to be λX B ⇒ {U𝒫 X|∀xU, ∃bB, x b b U} of type setsetset.
Proof:
Let X and B be given.
Assume HBasis.
We prove the intermediate claim HBleft: B 𝒫 X (∀xX, ∃bB, x b).
An exact proof term for the current goal is (andEL (B 𝒫 X (∀xX, ∃bB, x b)) (∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2) HBasis).
We prove the intermediate claim HBint: ∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2.
An exact proof term for the current goal is (andER (B 𝒫 X (∀xX, ∃bB, x b)) (∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2) HBasis).
We prove the intermediate claim HBsub: B 𝒫 X.
An exact proof term for the current goal is (andEL (B 𝒫 X) (∀xX, ∃bB, x b) HBleft).
We prove the intermediate claim HBcov: ∀xX, ∃bB, x b.
An exact proof term for the current goal is (andER (B 𝒫 X) (∀xX, ∃bB, x b) HBleft).
We prove the intermediate claim proofA: generated_topology X B 𝒫 X.
Let U be given.
Assume HU: U generated_topology X B.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : set∀xU0, ∃bB, x b b U0) U HU).
We prove the intermediate claim proofB: Empty generated_topology X B.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀xU0, ∃bB, x b b U0) Empty (Empty_In_Power X) (λx HxEmpty ⇒ EmptyE x HxEmpty (∃bB, x b b Empty))).
We prove the intermediate claim proofC: X generated_topology X B.
We prove the intermediate claim HXprop: ∀xX, ∃bB, x b b X.
Let x be given.
Assume HxX.
We prove the intermediate claim Hexb: ∃bB, x b.
An exact proof term for the current goal is (HBcov x HxX).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (andEL (b B) (x b) Hbpair).
We prove the intermediate claim Hxb: x b.
An exact proof term for the current goal is (andER (b B) (x b) Hbpair).
We prove the intermediate claim HbsubX: b X.
An exact proof term for the current goal is (PowerE X b (HBsub b HbB)).
We use b to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HbB.
Apply andI to the current goal.
An exact proof term for the current goal is Hxb.
An exact proof term for the current goal is HbsubX.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀xU0, ∃bB, x b b U0) X (Self_In_Power X) HXprop).
We prove the intermediate claim proofD: ∀UFam𝒫 (generated_topology X B), UFam generated_topology X B.
Let UFam be given.
Assume Hfam: UFam 𝒫 (generated_topology X B).
We prove the intermediate claim HsubFam: UFam generated_topology X B.
An exact proof term for the current goal is (PowerE (generated_topology X B) UFam Hfam).
We prove the intermediate claim HPowUnion: UFam 𝒫 X.
Apply PowerI X ( UFam) to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUtop: U generated_topology X B.
An exact proof term for the current goal is (HsubFam U HUin).
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) U HUtop)).
An exact proof term for the current goal is (HUsubX x HxU).
We prove the intermediate claim HUnionProp: ∀x UFam, ∃bB, x b b UFam.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUtop: U generated_topology X B.
An exact proof term for the current goal is (HsubFam U HUin).
We prove the intermediate claim HUprop: ∀x0U, ∃bB, x0 b b U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) U HUtop).
We prove the intermediate claim Hexb: ∃bB, x b b U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (andEL (b B) (x b b U) Hbpair).
We prove the intermediate claim Hbprop: x b b U.
An exact proof term for the current goal is (andER (b B) (x b b U) Hbpair).
We prove the intermediate claim Hxb: x b.
An exact proof term for the current goal is (andEL (x b) (b U) Hbprop).
We prove the intermediate claim HbSubU: b U.
An exact proof term for the current goal is (andER (x b) (b U) Hbprop).
We use b to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HbB.
Apply andI to the current goal.
An exact proof term for the current goal is Hxb.
Let y be given.
Assume Hyb.
Apply UnionI UFam y U (HbSubU y Hyb) HUin to the current goal.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀xU0, ∃bB, x b b U0) ( UFam) HPowUnion HUnionProp).
We prove the intermediate claim proofE: ∀Ugenerated_topology X B, ∀Vgenerated_topology X B, U V generated_topology X B.
Let U be given.
Assume HUtop.
Let V be given.
Assume HVtop.
We prove the intermediate claim HUprop: ∀x0U, ∃bB, x0 b b U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) U HUtop).
We prove the intermediate claim HVprop: ∀x0V, ∃bB, x0 b b V.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) V HVtop).
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) U HUtop)).
We prove the intermediate claim HPowCap: U V 𝒫 X.
Apply PowerI X (U V) to the current goal.
Let x be given.
Assume HxCap.
Apply binintersectE U V x HxCap to the current goal.
Assume HxU HxV.
An exact proof term for the current goal is (HUsubX x HxU).
We prove the intermediate claim HCapProp: ∀xU V, ∃bB, x b b U V.
Let x be given.
Assume HxCap.
Apply binintersectE U V x HxCap to the current goal.
Assume HxU HxV.
We prove the intermediate claim Hexb1: ∃b1B, x b1 b1 U.
An exact proof term for the current goal is (HUprop x HxU).
We prove the intermediate claim Hexb2: ∃b2B, x b2 b2 V.
An exact proof term for the current goal is (HVprop x HxV).
Apply Hexb1 to the current goal.
Let b1 be given.
Assume Hbpair1.
We prove the intermediate claim Hb1: b1 B.
An exact proof term for the current goal is (andEL (b1 B) (x b1 b1 U) Hbpair1).
We prove the intermediate claim Hb1prop: x b1 b1 U.
An exact proof term for the current goal is (andER (b1 B) (x b1 b1 U) Hbpair1).
We prove the intermediate claim Hb1x: x b1.
An exact proof term for the current goal is (andEL (x b1) (b1 U) Hb1prop).
We prove the intermediate claim Hb1Sub: b1 U.
An exact proof term for the current goal is (andER (x b1) (b1 U) Hb1prop).
Apply Hexb2 to the current goal.
Let b2 be given.
Assume Hbpair2.
We prove the intermediate claim Hb2: b2 B.
An exact proof term for the current goal is (andEL (b2 B) (x b2 b2 V) Hbpair2).
We prove the intermediate claim Hb2prop: x b2 b2 V.
An exact proof term for the current goal is (andER (b2 B) (x b2 b2 V) Hbpair2).
We prove the intermediate claim Hb2x: x b2.
An exact proof term for the current goal is (andEL (x b2) (b2 V) Hb2prop).
We prove the intermediate claim Hb2Sub: b2 V.
An exact proof term for the current goal is (andER (x b2) (b2 V) Hb2prop).
We prove the intermediate claim Hexb3: ∃b3B, x b3 b3 b1 b2.
An exact proof term for the current goal is (HBint b1 Hb1 b2 Hb2 x Hb1x Hb2x).
Apply Hexb3 to the current goal.
Let b3 be given.
Assume Hbpair3.
We prove the intermediate claim Hb3: b3 B.
An exact proof term for the current goal is (andEL (b3 B) (x b3 b3 b1 b2) Hbpair3).
We prove the intermediate claim Hb3prop: x b3 b3 b1 b2.
An exact proof term for the current goal is (andER (b3 B) (x b3 b3 b1 b2) Hbpair3).
We prove the intermediate claim HxB3: x b3.
An exact proof term for the current goal is (andEL (x b3) (b3 b1 b2) Hb3prop).
We prove the intermediate claim Hb3Sub: b3 b1 b2.
An exact proof term for the current goal is (andER (x b3) (b3 b1 b2) Hb3prop).
We use b3 to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hb3.
Apply andI to the current goal.
An exact proof term for the current goal is HxB3.
Let y be given.
Assume Hyb3.
We prove the intermediate claim Hy_in_b1b2: y b1 b2.
An exact proof term for the current goal is (Hb3Sub y Hyb3).
Apply binintersectE b1 b2 y Hy_in_b1b2 to the current goal.
Assume Hyb1 Hyb2.
Apply binintersectI U V y (Hb1Sub y Hyb1) (Hb2Sub y Hyb2) to the current goal.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) (U V) HPowCap HCapProp).
Apply andI to the current goal.
Apply andI to the current goal.
Apply andI to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is proofA.
An exact proof term for the current goal is proofB.
An exact proof term for the current goal is proofC.
An exact proof term for the current goal is proofD.
An exact proof term for the current goal is proofE.
Theorem. (generated_topology_contains_basis)
∀X B : set, basis_on X B∀b : set, b Bb generated_topology X B
Proof:
Let X and B be given.
Assume HBasis.
We prove the intermediate claim HBsub: B 𝒫 X.
An exact proof term for the current goal is (andEL (B 𝒫 X) (∀xX, ∃bB, x b) (andEL (B 𝒫 X (∀xX, ∃bB, x b)) (∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2) HBasis)).
We prove the intermediate claim HBint: ∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2.
An exact proof term for the current goal is (andER (B 𝒫 X (∀xX, ∃bB, x b)) (∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2) HBasis).
Let b0 be given.
Assume Hb0.
We prove the intermediate claim Hb0_subX: b0 X.
An exact proof term for the current goal is (PowerE X b0 (HBsub b0 Hb0)).
We prove the intermediate claim Hb0prop: ∀xb0, ∃bB, x b b b0.
Let x be given.
Assume Hxb0.
We prove the intermediate claim Hexb3: ∃b3B, x b3 b3 b0 b0.
An exact proof term for the current goal is (HBint b0 Hb0 b0 Hb0 x Hxb0 Hxb0).
Apply Hexb3 to the current goal.
Let b3 be given.
Assume Hb3pair.
We prove the intermediate claim Hb3: b3 B.
An exact proof term for the current goal is (andEL (b3 B) (x b3 b3 b0 b0) Hb3pair).
We prove the intermediate claim Hb3prop: x b3 b3 b0 b0.
An exact proof term for the current goal is (andER (b3 B) (x b3 b3 b0 b0) Hb3pair).
We prove the intermediate claim Hxb3: x b3.
An exact proof term for the current goal is (andEL (x b3) (b3 b0 b0) Hb3prop).
We prove the intermediate claim Hb3sub_inter: b3 b0 b0.
An exact proof term for the current goal is (andER (x b3) (b3 b0 b0) Hb3prop).
We prove the intermediate claim Hb3subb0: b3 b0.
Let y be given.
Assume Hyb3.
We prove the intermediate claim Hycap: y b0 b0.
An exact proof term for the current goal is (Hb3sub_inter y Hyb3).
Apply binintersectE b0 b0 y Hycap to the current goal.
Assume Hy1 Hy2.
An exact proof term for the current goal is Hy1.
We use b3 to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hb3.
Apply andI to the current goal.
An exact proof term for the current goal is Hxb3.
An exact proof term for the current goal is Hb3subb0.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀xU0, ∃bB, x b b U0) b0 (PowerI X b0 Hb0_subX) Hb0prop).
Definition. We define basis_generates to be λX B T ⇒ basis_on X B generated_topology X B = T of type setsetsetprop.
Definition. We define basis_refines to be λX B T ⇒ topology_on X T (∀UT, ∀xU, ∃bB, x b b U) of type setsetsetprop.
Proof:
Let X and B be given.
Assume HBasis.
Use reflexivity.
Theorem. (open_sets_as_unions_of_basis)
∀X B : set, basis_on X B∀U : set, open_in X (generated_topology X B) U∃Fam𝒫 B, Fam = U
Proof:
Let X and B be given.
Assume HBasis.
We prove the intermediate claim HBsub: B 𝒫 X.
An exact proof term for the current goal is (andEL (B 𝒫 X) (∀xX, ∃bB, x b) (andEL (B 𝒫 X (∀xX, ∃bB, x b)) (∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2) HBasis)).
Let U be given.
Assume HUopen.
We prove the intermediate claim HUtop: U generated_topology X B.
An exact proof term for the current goal is (andER (topology_on X (generated_topology X B)) (U generated_topology X B) HUopen).
We prove the intermediate claim HUprop: ∀xU, ∃bB, x b b U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) U HUtop).
Set Fam to be the term {bB|b U} of type set.
We prove the intermediate claim HFamPow: Fam 𝒫 B.
Apply PowerI B Fam to the current goal.
Let b be given.
Assume HbFam.
An exact proof term for the current goal is (SepE1 B (λb0 : setb0 U) b HbFam).
We prove the intermediate claim HUnion_eq: Fam = U.
Apply set_ext to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred Fam x HxUnion to the current goal.
Let b be given.
Assume Hxb HbFam.
We prove the intermediate claim HbsubU: b U.
An exact proof term for the current goal is (SepE2 B (λb0 : setb0 U) b HbFam).
An exact proof term for the current goal is (HbsubU x Hxb).
Let x be given.
Assume HxU.
We prove the intermediate claim Hexb: ∃bB, x b b U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (andEL (b B) (x b b U) Hbpair).
We prove the intermediate claim Hbprop: x b b U.
An exact proof term for the current goal is (andER (b B) (x b b U) Hbpair).
We prove the intermediate claim Hxb: x b.
An exact proof term for the current goal is (andEL (x b) (b U) Hbprop).
We prove the intermediate claim HbsubU: b U.
An exact proof term for the current goal is (andER (x b) (b U) Hbprop).
We prove the intermediate claim HbFam: b Fam.
An exact proof term for the current goal is (SepI B (λb0 : setb0 U) b HbB HbsubU).
An exact proof term for the current goal is (UnionI Fam x b Hxb HbFam).
We use Fam to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HFamPow.
An exact proof term for the current goal is HUnion_eq.
Theorem. (basis_generates_open_sets)
∀X B : set, basis_on X B∀U : set, (∃Fam𝒫 B, Fam = U)open_in X (generated_topology X B) U
Proof:
Let X and B be given.
Assume HBasis.
We prove the intermediate claim HBsub: B 𝒫 X.
An exact proof term for the current goal is (andEL (B 𝒫 X) (∀xX, ∃bB, x b) (andEL (B 𝒫 X (∀xX, ∃bB, x b)) (∀b1B, ∀b2B, ∀x : set, x b1x b2∃b3B, x b3 b3 b1 b2) HBasis)).
Let U be given.
Assume Hex.
We prove the intermediate claim HUGen: U generated_topology X B.
Apply Hex to the current goal.
Let Fam be given.
Assume HFampair.
We prove the intermediate claim HFamPow: Fam 𝒫 B.
An exact proof term for the current goal is (andEL (Fam 𝒫 B) ( Fam = U) HFampair).
We prove the intermediate claim HUnionEq: Fam = U.
An exact proof term for the current goal is (andER (Fam 𝒫 B) ( Fam = U) HFampair).
We prove the intermediate claim HFamSubB: Fam B.
An exact proof term for the current goal is (PowerE B Fam HFamPow).
We prove the intermediate claim HFamSubX: Fam 𝒫 X.
Let b be given.
Assume HbFam.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (HFamSubB b HbFam).
An exact proof term for the current goal is (HBsub b HbB).
We prove the intermediate claim HUnionSubX: Fam X.
Let x be given.
Assume HxUnion.
Apply UnionE_impred Fam x HxUnion to the current goal.
Let b be given.
Assume Hxb HbFam.
We prove the intermediate claim HbSubX: b X.
An exact proof term for the current goal is (PowerE X b (HFamSubX b HbFam)).
An exact proof term for the current goal is (HbSubX x Hxb).
We prove the intermediate claim HUnionSubU: Fam U.
rewrite the current goal using HUnionEq (from left to right).
An exact proof term for the current goal is (Subq_ref U).
We prove the intermediate claim HUsubUnion: U Fam.
rewrite the current goal using HUnionEq (from right to left).
An exact proof term for the current goal is (Subq_ref ( Fam)).
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (Subq_tra U ( Fam) X HUsubUnion HUnionSubX).
We prove the intermediate claim HUpropU: ∀xU, ∃bB, x b b U.
Let x be given.
Assume HxU.
We prove the intermediate claim HxUnion: x Fam.
An exact proof term for the current goal is (HUsubUnion x HxU).
Apply UnionE_impred Fam x HxUnion to the current goal.
Let b be given.
Assume Hxb HbFam.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (HFamSubB b HbFam).
We prove the intermediate claim HbSubUnion: b Fam.
Let y be given.
Assume Hyb.
An exact proof term for the current goal is (UnionI Fam y b Hyb HbFam).
We prove the intermediate claim HbSubU: b U.
An exact proof term for the current goal is (Subq_tra b ( Fam) U HbSubUnion HUnionSubU).
We use b to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HbB.
Apply andI to the current goal.
An exact proof term for the current goal is Hxb.
An exact proof term for the current goal is HbSubU.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀x0U0, ∃b0B, x0 b0 b0 U0) U (PowerI X U HUsubX) HUpropU).
An exact proof term for the current goal is (andI (topology_on X (generated_topology X B)) (U generated_topology X B) (lemma_topology_from_basis X B HBasis) HUGen).
Theorem. (open_as_union_of_basis_elements)
∀X B : set, basis_on X B∀U : set, open_in X (generated_topology X B) UU = {bB|b U}
Proof:
Let X and B be given.
Assume HBasis.
Let U be given.
Assume HUopen.
We prove the intermediate claim HUtop: U generated_topology X B.
An exact proof term for the current goal is (andER (topology_on X (generated_topology X B)) (U generated_topology X B) HUopen).
We prove the intermediate claim HUprop: ∀xU, ∃bB, x b b U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) U HUtop).
Set Fam to be the term {bB|b U} of type set.
Apply set_ext to the current goal.
Let x be given.
Assume HxU.
We prove the intermediate claim Hexb: ∃bB, x b b U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (andEL (b B) (x b b U) Hbpair).
We prove the intermediate claim Hbprop: x b b U.
An exact proof term for the current goal is (andER (b B) (x b b U) Hbpair).
We prove the intermediate claim Hxb: x b.
An exact proof term for the current goal is (andEL (x b) (b U) Hbprop).
We prove the intermediate claim HbsubU: b U.
An exact proof term for the current goal is (andER (x b) (b U) Hbprop).
We prove the intermediate claim HbFam: b Fam.
An exact proof term for the current goal is (SepI B (λb0 : setb0 U) b HbB HbsubU).
An exact proof term for the current goal is (UnionI Fam x b Hxb HbFam).
Let x be given.
Assume HxUnion.
Apply UnionE_impred Fam x HxUnion to the current goal.
Let b be given.
Assume Hxb HbFam.
We prove the intermediate claim HbsubU: b U.
An exact proof term for the current goal is (SepE2 B (λb0 : setb0 U) b HbFam).
An exact proof term for the current goal is (HbsubU x Hxb).
Theorem. (basis_refines_topology)
∀X T C : set, topology_on X T(∀cC, c T)(∀UT, ∀xU, ∃CxC, x Cx Cx U)basis_on X C generated_topology X C = T
Proof:
Let X, T and C be given.
Assume Htop HCsub Href.
We prove the intermediate claim Hleft: ((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T).
An exact proof term for the current goal is (andEL (((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T)) (∀UT, ∀VT, U V T) Htop).
We prove the intermediate claim Hcore: (T 𝒫 X Empty T) X T.
An exact proof term for the current goal is (andEL ((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T) Hleft).
We prove the intermediate claim HTPowEmpty: T 𝒫 X Empty T.
An exact proof term for the current goal is (andEL (T 𝒫 X Empty T) (X T) Hcore).
We prove the intermediate claim HTsubPow: T 𝒫 X.
An exact proof term for the current goal is (andEL (T 𝒫 X) (Empty T) HTPowEmpty).
We prove the intermediate claim HXT: X T.
An exact proof term for the current goal is (andER (T 𝒫 X Empty T) (X T) Hcore).
We prove the intermediate claim HUnionClosed: ∀UFam𝒫 T, UFam T.
An exact proof term for the current goal is (andER ((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T) Hleft).
We prove the intermediate claim HInterClosed: ∀UT, ∀VT, U V T.
An exact proof term for the current goal is (andER (((T 𝒫 X Empty T) X T) (∀UFam𝒫 T, UFam T)) (∀UT, ∀VT, U V T) Htop).
We prove the intermediate claim HBasis: basis_on X C.
We will prove (C 𝒫 X (∀xX, ∃cC, x c) (∀b1C, ∀b2C, ∀x : set, x b1x b2∃b3C, x b3 b3 b1 b2)).
Apply andI to the current goal.
Apply andI to the current goal.
Let c be given.
Assume HcC.
An exact proof term for the current goal is (HTsubPow c (HCsub c HcC)).
Let x be given.
Assume HxX.
We prove the intermediate claim Hex: ∃cC, x c c X.
An exact proof term for the current goal is (Href X HXT x HxX).
Apply Hex to the current goal.
Let c be given.
Assume Hpair.
We prove the intermediate claim HcC: c C.
An exact proof term for the current goal is (andEL (c C) (x c c X) Hpair).
We prove the intermediate claim Hcprop: x c c X.
An exact proof term for the current goal is (andER (c C) (x c c X) Hpair).
We prove the intermediate claim Hxc: x c.
An exact proof term for the current goal is (andEL (x c) (c X) Hcprop).
We use c to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HcC.
An exact proof term for the current goal is Hxc.
Let c1 be given.
Assume Hc1C.
Let c2 be given.
Assume Hc2C.
Let x be given.
Assume Hxc1 Hxc2.
We prove the intermediate claim Hc1T: c1 T.
An exact proof term for the current goal is (HCsub c1 Hc1C).
We prove the intermediate claim Hc2T: c2 T.
An exact proof term for the current goal is (HCsub c2 Hc2C).
We prove the intermediate claim HcapT: c1 c2 T.
An exact proof term for the current goal is (HInterClosed c1 Hc1T c2 Hc2T).
We prove the intermediate claim HxCap: x c1 c2.
An exact proof term for the current goal is (binintersectI c1 c2 x Hxc1 Hxc2).
We prove the intermediate claim Hex: ∃c3C, x c3 c3 c1 c2.
An exact proof term for the current goal is (Href (c1 c2) HcapT x HxCap).
An exact proof term for the current goal is Hex.
We prove the intermediate claim Hgen_sub_T: generated_topology X C T.
Let U be given.
Assume HUgen: U generated_topology X C.
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set∀x0U0, ∃b0C, x0 b0 b0 U0) U HUgen)).
We prove the intermediate claim HUprop: ∀xU, ∃cC, x c c U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃b0C, x0 b0 b0 U0) U HUgen).
Set Fam to be the term {cC|c U} of type set.
We prove the intermediate claim HFamSubC: Fam C.
Let c be given.
Assume HcFam.
An exact proof term for the current goal is (SepE1 C (λc0 : setc0 U) c HcFam).
We prove the intermediate claim HFamSubT: Fam T.
Let c be given.
Assume HcFam.
We prove the intermediate claim HcC: c C.
An exact proof term for the current goal is (HFamSubC c HcFam).
An exact proof term for the current goal is (HCsub c HcC).
We prove the intermediate claim HFamPowT: Fam 𝒫 T.
An exact proof term for the current goal is (PowerI T Fam HFamSubT).
We prove the intermediate claim HUnionSubU: Fam U.
Let x be given.
Assume HxUnion.
Apply UnionE_impred Fam x HxUnion to the current goal.
Let c be given.
Assume Hxc HcFam.
We prove the intermediate claim Hcprop: c U.
An exact proof term for the current goal is (SepE2 C (λc0 : setc0 U) c HcFam).
An exact proof term for the current goal is (Hcprop x Hxc).
We prove the intermediate claim HUsubUnion: U Fam.
Let x be given.
Assume HxU.
We prove the intermediate claim Hex: ∃cC, x c c U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hex to the current goal.
Let c be given.
Assume Hcpair.
We prove the intermediate claim HcC: c C.
An exact proof term for the current goal is (andEL (c C) (x c c U) Hcpair).
We prove the intermediate claim Hcprop: x c c U.
An exact proof term for the current goal is (andER (c C) (x c c U) Hcpair).
We prove the intermediate claim Hxc: x c.
An exact proof term for the current goal is (andEL (x c) (c U) Hcprop).
We prove the intermediate claim HcsubU: c U.
An exact proof term for the current goal is (andER (x c) (c U) Hcprop).
We prove the intermediate claim HcFam: c Fam.
An exact proof term for the current goal is (SepI C (λc0 : setc0 U) c HcC HcsubU).
An exact proof term for the current goal is (UnionI Fam x c Hxc HcFam).
We prove the intermediate claim HUnionEqU: Fam = U.
Apply set_ext to the current goal.
Let x be given.
Assume HxUnion.
An exact proof term for the current goal is (HUnionSubU x HxUnion).
Let x be given.
Assume HxU.
An exact proof term for the current goal is (HUsubUnion x HxU).
We prove the intermediate claim HUnionInT: Fam T.
An exact proof term for the current goal is (HUnionClosed Fam HFamPowT).
rewrite the current goal using HUnionEqU (from right to left).
An exact proof term for the current goal is HUnionInT.
We prove the intermediate claim HT_sub_gen: T generated_topology X C.
Let U be given.
Assume HU: U T.
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U (HTsubPow U HU)).
We prove the intermediate claim HUprop: ∀xU, ∃cC, x c c U.
Let x be given.
Assume HxU.
An exact proof term for the current goal is (Href U HU x HxU).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀x0U0, ∃b0C, x0 b0 b0 U0) U (PowerI X U HUsubX) HUprop).
We prove the intermediate claim HEqual: generated_topology X C = T.
Apply set_ext to the current goal.
Let U be given.
Assume HUgen.
An exact proof term for the current goal is (Hgen_sub_T U HUgen).
Let U be given.
Assume HU.
An exact proof term for the current goal is (HT_sub_gen U HU).
Apply andI to the current goal.
An exact proof term for the current goal is HBasis.
An exact proof term for the current goal is HEqual.
Theorem. (lemma13_2_basis_from_open_subcollection)
∀X T C : set, topology_on X T(∀cC, c T)(∀UT, ∀xU, ∃cC, x c c U)basis_on X C generated_topology X C = T
Proof:
Let X, T and C be given.
Assume Htop HCsub Href.
An exact proof term for the current goal is (basis_refines_topology X T C Htop HCsub Href).
Theorem. (finer_via_basis)
∀X B B' : set, basis_on X Bbasis_on X B'(∀xX, ∀b : set, b Bx b∃b'B', x b' b' b)finer_than (generated_topology X B') (generated_topology X B)
Proof:
Let X, B and B' be given.
Assume HB HB' Hcond.
We prove the intermediate claim HT: topology_on X (generated_topology X B).
An exact proof term for the current goal is (lemma_topology_from_basis X B HB).
We prove the intermediate claim HRefProp: ∀Ugenerated_topology X B, ∀xU, ∃b'B', x b' b' U.
Let U be given.
Assume HU: U generated_topology X B.
Let x be given.
Assume HxU.
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) U HU)).
We prove the intermediate claim HxX: x X.
An exact proof term for the current goal is (HUsubX x HxU).
We prove the intermediate claim HUprop: ∀x0U, ∃bB, x0 b b U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃bB, x0 b b U0) U HU).
We prove the intermediate claim Hexb: ∃bB, x b b U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (andEL (b B) (x b b U) Hbpair).
We prove the intermediate claim Hbprop: x b b U.
An exact proof term for the current goal is (andER (b B) (x b b U) Hbpair).
We prove the intermediate claim Hxb: x b.
An exact proof term for the current goal is (andEL (x b) (b U) Hbprop).
We prove the intermediate claim HbsubU: b U.
An exact proof term for the current goal is (andER (x b) (b U) Hbprop).
We prove the intermediate claim Hexb': ∃b'B', x b' b' b.
An exact proof term for the current goal is (Hcond x HxX b HbB Hxb).
Apply Hexb' to the current goal.
Let b' be given.
Assume Hb'pair.
We prove the intermediate claim Hb'B: b' B'.
An exact proof term for the current goal is (andEL (b' B') (x b' b' b) Hb'pair).
We prove the intermediate claim Hb'prop: x b' b' b.
An exact proof term for the current goal is (andER (b' B') (x b' b' b) Hb'pair).
We prove the intermediate claim Hxb': x b'.
An exact proof term for the current goal is (andEL (x b') (b' b) Hb'prop).
We prove the intermediate claim Hb'subb: b' b.
An exact proof term for the current goal is (andER (x b') (b' b) Hb'prop).
We use b' to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hb'B.
Apply andI to the current goal.
An exact proof term for the current goal is Hxb'.
An exact proof term for the current goal is (Subq_tra b' b U Hb'subb HbsubU).
We will prove generated_topology X B generated_topology X B'.
Let U be given.
Assume HU.
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set∀x0U0, ∃b0B, x0 b0 b0 U0) U HU)).
We prove the intermediate claim HUprop: ∀xU, ∃b'B', x b' b' U.
An exact proof term for the current goal is (HRefProp U HU).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀x0U0, ∃b0B', x0 b0 b0 U0) U (PowerI X U HUsubX) HUprop).
Theorem. (basis_finer_equiv_condition)
∀X B B' : set, basis_on X Bbasis_on X B'((∀xX, ∀bB, x b∃b'B', x b' b' b) finer_than (generated_topology X B') (generated_topology X B))
Proof:
Let X, B and B' be given.
Assume HB HB'.
Apply iffI to the current goal.
Assume Hcond.
An exact proof term for the current goal is (finer_via_basis X B B' HB HB' Hcond).
Assume Hfiner.
Let x be given.
Assume HxX.
Let b be given.
Assume HbB Hxb.
We prove the intermediate claim HbGen: b generated_topology X B.
An exact proof term for the current goal is (generated_topology_contains_basis X B HB b HbB).
We prove the intermediate claim HbGen': b generated_topology X B'.
An exact proof term for the current goal is (Hfiner b HbGen).
We prove the intermediate claim Hbprop: ∀x0b, ∃b'B', x0 b' b' b.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃b0B', x0 b0 b0 U0) b HbGen').
An exact proof term for the current goal is (Hbprop x Hxb).
Theorem. (generated_topology_finer)
∀X B T : set, basis_on X Btopology_on X T(∀bB, b T)finer_than T (generated_topology X B)
Proof:
Let X, B and T be given.
Assume HBasis HT HBsub.
We prove the intermediate claim HUnionClosed: ∀Fam𝒫 T, Fam T.
An exact proof term for the current goal is (andER ((T 𝒫 X Empty T) X T) (∀Fam𝒫 T, Fam T) (andEL (((T 𝒫 X Empty T) X T) (∀Fam𝒫 T, Fam T)) (∀UT, ∀VT, U V T) HT)).
We will prove generated_topology X B T.
Let U be given.
Assume HU.
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set∀x0U0, ∃b0B, x0 b0 b0 U0) U HU)).
We prove the intermediate claim HUprop: ∀xU, ∃bB, x b b U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set∀x0U0, ∃b0B, x0 b0 b0 U0) U HU).
Set Fam to be the term {bB|b U} of type set.
We prove the intermediate claim HFamPowB: Fam 𝒫 B.
Apply PowerI B Fam to the current goal.
Let b be given.
Assume HbFam.
An exact proof term for the current goal is (SepE1 B (λb0 : setb0 U) b HbFam).
We prove the intermediate claim HUnionEq: Fam = U.
Apply set_ext to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred Fam x HxUnion to the current goal.
Let b be given.
Assume Hxb HbFam.
We prove the intermediate claim HbsubU: b U.
An exact proof term for the current goal is (SepE2 B (λb0 : setb0 U) b HbFam).
An exact proof term for the current goal is (HbsubU x Hxb).
Let x be given.
Assume HxU.
We prove the intermediate claim Hexb: ∃bB, x b b U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (andEL (b B) (x b b U) Hbpair).
We prove the intermediate claim Hbprop: x b b U.
An exact proof term for the current goal is (andER (b B) (x b b U) Hbpair).
We prove the intermediate claim Hxb: x b.
An exact proof term for the current goal is (andEL (x b) (b U) Hbprop).
We prove the intermediate claim HbsubU: b U.
An exact proof term for the current goal is (andER (x b) (b U) Hbprop).
We prove the intermediate claim HbT: b T.
An exact proof term for the current goal is (HBsub b HbB).
We prove the intermediate claim HbFam: b Fam.
An exact proof term for the current goal is (SepI B (λb0 : setb0 U) b HbB HbsubU).
An exact proof term for the current goal is (UnionI Fam x b Hxb HbFam).
We prove the intermediate claim HFamPowT: Fam 𝒫 T.
Apply PowerI T Fam to the current goal.
Let b be given.
Assume HbFam.
We prove the intermediate claim HbB: b B.
An exact proof term for the current goal is (SepE1 B (λb0 : setb0 U) b HbFam).
An exact proof term for the current goal is (HBsub b HbB).
We prove the intermediate claim HUnionT: Fam T.
An exact proof term for the current goal is (HUnionClosed Fam HFamPowT).
rewrite the current goal using HUnionEq (from right to left).
An exact proof term for the current goal is HUnionT.
Proof:
Let X, B and T be given.
Assume HBasis HT HBsub.
An exact proof term for the current goal is (generated_topology_finer X B T HBasis HT HBsub).
Proof:
Let X and B be given.
Assume HBasis.
Apply set_ext to the current goal.
Let U be given.
Assume HU.
We prove the intermediate claim HUopen: open_in X (generated_topology X B) U.
An exact proof term for the current goal is (andI (topology_on X (generated_topology X B)) (U generated_topology X B) (lemma_topology_from_basis X B HBasis) HU).
We prove the intermediate claim HexFam: ∃Fam𝒫 B, Fam = U.
An exact proof term for the current goal is (open_sets_as_unions_of_basis X B HBasis U HUopen).
Apply HexFam to the current goal.
Let Fam be given.
Assume HFampair.
We prove the intermediate claim HFamPow: Fam 𝒫 B.
An exact proof term for the current goal is (andEL (Fam 𝒫 B) ( Fam = U) HFampair).
We prove the intermediate claim HUnion: Fam = U.
An exact proof term for the current goal is (andER (Fam 𝒫 B) ( Fam = U) HFampair).
We prove the intermediate claim HUnionFam: Fam { Fam0|Fam0𝒫 B}.
An exact proof term for the current goal is (ReplI (𝒫 B) (λFam0 : set Fam0) Fam HFamPow).
rewrite the current goal using HUnion (from right to left).
An exact proof term for the current goal is HUnionFam.
Let U be given.
Assume HUUnion.
We prove the intermediate claim HexFamPowRaw: ∃Fam𝒫 B, U = Fam.
An exact proof term for the current goal is (ReplE (𝒫 B) (λFam0 : set Fam0) U HUUnion).
We prove the intermediate claim HexFamPow: ∃Fam𝒫 B, Fam = U.
Apply HexFamPowRaw to the current goal.
Let Fam be given.
Assume HFamPair.
We prove the intermediate claim HFamPow: Fam 𝒫 B.
An exact proof term for the current goal is (andEL (Fam 𝒫 B) (U = Fam) HFamPair).
We prove the intermediate claim HUnion: U = Fam.
An exact proof term for the current goal is (andER (Fam 𝒫 B) (U = Fam) HFamPair).
We use Fam to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HFamPow.
rewrite the current goal using HUnion (from right to left).
Use reflexivity.
We prove the intermediate claim HUopen: open_in X (generated_topology X B) U.
An exact proof term for the current goal is (basis_generates_open_sets X B HBasis U HexFamPow).
An exact proof term for the current goal is (andER (topology_on X (generated_topology X B)) (U generated_topology X B) HUopen).
Definition. We define singleton_basis to be λX ⇒ {{x,x}|xX} of type setset.
Proof:
The rest of this subproof is missing.
Proof:
Let X be given.
Apply set_ext to the current goal.
Let U be given.
Assume HUgen.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : set∀x0U0, ∃bsingleton_basis X, x0 b b U0) U HUgen).
Let U be given.
Assume HUinPow: U 𝒫 X.
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U HUinPow).
We prove the intermediate claim HUprop: ∀xU, ∃bsingleton_basis X, x b b U.
Let x be given.
Assume HxU.
We use {x,x} to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is (ReplI X (λx0 : set{x0,x0}) x (HUsubX x HxU)).
Apply andI to the current goal.
An exact proof term for the current goal is (UPairI1 x x).
Let y be given.
Assume Hy.
Apply (UPairE y x x Hy (y U)) to the current goal.
Assume Hyx.
rewrite the current goal using Hyx (from left to right).
An exact proof term for the current goal is HxU.
Assume Hyx.
rewrite the current goal using Hyx (from left to right).
An exact proof term for the current goal is HxU.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀x0U0, ∃bsingleton_basis X, x0 b b U0) U HUinPow HUprop).
Definition. We define OrderedPair to be λx y ⇒ UPair x (UPair x y) of type setsetset.
Definition. We define R to be real of type set.
Definition. We define Rlt to be λa b ⇒ a R b R a < b of type setsetprop.
Definition. We define EuclidPlane to be OrderedPair R R of type set.
Definition. We define distance_R2 to be λp c ⇒ Eps_i (λr ⇒ r R) of type setsetset.
Definition. We define circular_regions to be {U𝒫 EuclidPlane|∃c : set, ∃r : set, c EuclidPlane r R ¬ (r = 0) U = {pEuclidPlane|Rlt (distance_R2 p c) r}} of type set.
Definition. We define rectangular_regions to be {U𝒫 EuclidPlane|∃a b c d : set, a R b R c R d R U = {pEuclidPlane|∃x y : set, p = OrderedPair x y Rlt a x Rlt x b Rlt c y Rlt y d}} of type set.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
Let X, B and B' be given.
Assume HBasis Href.
We prove the intermediate claim Hprop: ∀Ugenerated_topology X B, ∀xU, ∃b'B', x b' b' U.
An exact proof term for the current goal is (andER (topology_on X (generated_topology X B)) (∀Ugenerated_topology X B, ∀xU, ∃b'B', x b' b' U) Href).
We will prove generated_topology X B generated_topology X B'.
Let U be given.
Assume HU.
We prove the intermediate claim HUsubX: U X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set∀x0U0, ∃b0B, x0 b0 b0 U0) U HU)).
We prove the intermediate claim HUprop: ∀xU, ∃b'B', x b' b' U.
An exact proof term for the current goal is (Hprop U HU).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set∀x0U0, ∃b0B', x0 b0 b0 U0) U (PowerI X U HUsubX) HUprop).
Definition. We define subbasis_on to be λX S ⇒ S 𝒫 X of type setsetprop.
Definition. We define intersection_of_family to be λFam ⇒ {x Fam|∀U : set, U Famx U} of type setset.
Definition. We define finite_subcollections to be λS ⇒ {F𝒫 S|finite F} of type setset.
Definition. We define finite_intersections_of to be λS ⇒ {intersection_of_family F|Ffinite_subcollections S} of type setset.
Definition. We define basis_of_subbasis to be λ_ S ⇒ {bfinite_intersections_of S|b Empty} of type setsetset.
Definition. We define generated_topology_from_subbasis to be λX S ⇒ generated_topology X (basis_of_subbasis X S) of type setsetset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (ex13_1_local_open_subset)
∀X T A : set, topology_on X T(∀xA, ∃UT, x U U A)open_in X T A
Proof:
The rest of this subproof is missing.
Definition. We define a_elt to be Empty of type set.
Definition. We define b_elt to be {Empty} of type set.
Definition. We define c_elt to be {{Empty}} of type set.
Definition. We define abc_set to be UPair a_elt (UPair b_elt c_elt) of type set.
Definition. We define top_abc_1 to be UPair Empty abc_set of type set.
Definition. We define top_abc_2 to be 𝒫 abc_set of type set.
Definition. We define top_abc_3 to be UPair Empty (UPair {a_elt} abc_set) of type set.
Definition. We define top_abc_4 to be UPair Empty (UPair {b_elt} abc_set) of type set.
Definition. We define top_abc_5 to be UPair Empty (UPair {c_elt} abc_set) of type set.
Definition. We define top_abc_6 to be UPair Empty (UPair (UPair a_elt b_elt) abc_set) of type set.
Definition. We define top_abc_7 to be UPair Empty (UPair (UPair a_elt c_elt) abc_set) of type set.
Definition. We define top_abc_8 to be UPair Empty (UPair (UPair b_elt c_elt) abc_set) of type set.
Definition. We define top_abc_9 to be UPair Empty (UPair {a_elt} (UPair (UPair a_elt b_elt) abc_set)) of type set.
Proof:
The rest of this subproof is missing.
Definition. We define Intersection_Fam to be λFam ⇒ {U𝒫 ( Fam)|∀T : set, T FamU T} of type setset.
Definition. We define infinite_complement_family to be λX ⇒ {U𝒫 X|infinite (X U) U = Empty U = X} of type setset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (ex13_4b_smallest_largest)
∀X Fam : set, ∃Tmin, topology_on X Tmin (∀TFam, T Tmin) (∀T', topology_on X T' (∀TFam, T T')Tmin T') ∃Tmax, topology_on X Tmax (∀TFam, Tmax T) (∀T', topology_on X T' (∀TFam, T' T)T' Tmax)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define rational_numbers to be ω of type set.
Definition. We define open_interval to be λa b ⇒ {xR|Rlt a x Rlt x b} of type setsetset.
Definition. We define halfopen_interval_left to be λa b ⇒ {xR|Rlt a x ¬ (Rlt b x)} of type setsetset.
Definition. We define R_standard_basis to be aR{open_interval a b|bR} of type set.
Definition. We define R_standard_topology to be generated_topology R R_standard_basis of type set.
Definition. We define R_lower_limit_basis to be aR{halfopen_interval_left a b|bR} of type set.
Definition. We define R_lower_limit_topology to be generated_topology R R_lower_limit_basis of type set.
Definition. We define K_set to be ω of type set.
Definition. We define R_K_basis to be aR{open_interval a b K_set|bR} of type set.
Definition. We define R_K_topology to be generated_topology R (R_standard_basis R_K_basis) of type set.
Proof:
The rest of this subproof is missing.
Definition. We define R_finite_complement_topology to be countable_complement_topology R of type set.
Definition. We define R_upper_limit_topology to be R_lower_limit_topology of type set.
Definition. We define R_ray_topology to be {U𝒫 R|U = Empty U = R (∃aR, {xR|Rlt a x} U)} of type set.
Proof:
The rest of this subproof is missing.
Definition. We define rational_open_intervals_basis to be q1rational_numbers{open_interval q1 q2|q2rational_numbers} of type set.
Proof:
The rest of this subproof is missing.
Definition. We define rational_halfopen_intervals_basis to be q1rational_numbers{halfopen_interval_left q1 q2|q2rational_numbers} of type set.
Proof:
The rest of this subproof is missing.
Definition. We define order_topology_basis to be λX ⇒ 𝒫 X of type setset.
Definition. We define order_topology to be λX ⇒ generated_topology X (order_topology_basis X) of type setset.
Proof:
The rest of this subproof is missing.
Definition. We define open_ray_upper to be λX a ⇒ X of type setsetset.
Definition. We define open_ray_lower to be λX a ⇒ X of type setsetset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define R2_dictionary_order_topology to be order_topology (OrderedPair R R) of type set.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define Zplus to be ω of type set.
Proof:
The rest of this subproof is missing.
Definition. We define two_by_nat to be OrderedPair 2 ω of type set.
Definition. We define two_by_nat_order_topology to be order_topology two_by_nat of type set.
Proof:
The rest of this subproof is missing.
Definition. We define rectangle_set to be λU V ⇒ OrderedPair U V of type setsetset.
Definition. We define product_subbasis to be λX Tx Y Ty ⇒ UTx{rectangle_set U V|VTy} of type setsetsetsetset.
Definition. We define product_topology to be λX Tx Y Ty ⇒ generated_topology (OrderedPair X Y) (product_subbasis X Tx Y Ty) of type setsetsetsetset.
Proof:
The rest of this subproof is missing.
Theorem. (product_basis_generates)
∀X Tx Y Ty Bx By : set, basis_on X Bx generated_topology X Bx = Txbasis_on Y By generated_topology Y By = Ty∃B : set, basis_on (OrderedPair X Y) B (∀UBx, ∀VBy, OrderedPair U V B) generated_topology (OrderedPair X Y) B = product_topology X Tx Y Ty
Proof:
The rest of this subproof is missing.
Definition. We define projection1 to be λX Y ⇒ {p𝒫 (OrderedPair (OrderedPair X Y) X)|∃x : set, ∃y : set, x X y Y p = UPair (OrderedPair x y) x} of type setsetset.
Definition. We define projection2 to be λX Y ⇒ {p𝒫 (OrderedPair (OrderedPair X Y) Y)|∃x : set, ∃y : set, x X y Y p = UPair (OrderedPair x y) y} of type setsetset.
Proof:
The rest of this subproof is missing.
Definition. We define apply_fun to be λf x ⇒ Eps_i (λy ⇒ UPair x y f) of type setsetset.
Definition. We define function_on to be λf X Y ⇒ ∀x : set, x Xapply_fun f x Y of type setsetsetprop.
Definition. We define function_space to be λX Y ⇒ {f𝒫 (OrderedPair X Y)|function_on f X Y} of type setsetset.
Definition. We define const_family to be λI X ⇒ {UPair i X|iI} of type setsetset.
Definition. We define product_component to be λXi i ⇒ apply_fun Xi i of type setsetset.
Definition. We define product_component_topology to be λXi i ⇒ apply_fun Xi i of type setsetset.
Definition. We define product_space to be λI Xi ⇒ {f𝒫 ( Xi)|function_on f I ( Xi) ∀i : set, i Iapply_fun f i apply_fun Xi i} of type setsetset.
Definition. We define product_topology_full to be λI Xi ⇒ generated_topology (product_space I Xi) Empty of type setsetset.
Definition. We define box_topology to be λI Xi ⇒ product_topology_full I Xi of type setsetset.
Definition. We define countable_product_space to be λI Xi ⇒ product_space I Xi of type setsetset.
Definition. We define countable_product_topology to be λI Xi ⇒ product_topology_full I Xi of type setsetset.
Definition. We define euclidean_space to be λn ⇒ product_space n (const_family n R) of type setset.
Definition. We define euclidean_topology to be λn ⇒ product_topology_full n (const_family n R) of type setset.
Definition. We define R2_standard_topology to be product_topology R R_standard_topology R R_standard_topology of type set.
Proof:
The rest of this subproof is missing.
Definition. We define subspace_topology to be λX Tx Y ⇒ {U𝒫 Y|∃VTx, U = V Y} of type setsetsetset.
Proof:
The rest of this subproof is missing.
Theorem. (open_in_subspace_iff)
∀X Tx Y U : set, topology_on X TxY Xopen_in Y (subspace_topology X Tx Y) U ∃VTx, U = V Y
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (open_in_subspace_if_ambient_open)
∀X Tx Y U : set, topology_on X TxY Txopen_in Y (subspace_topology X Tx Y) UU Tx
Proof:
The rest of this subproof is missing.
Theorem. (product_subspace_topology)
∀X Tx Y Ty A B : set, topology_on X Txtopology_on Y TyA XB Yproduct_topology A (subspace_topology X Tx A) B (subspace_topology Y Ty B) = subspace_topology (OrderedPair X Y) (product_topology X Tx Y Ty) (OrderedPair A B)
Proof:
The rest of this subproof is missing.
Definition. We define unit_interval to be R of type set.
Definition. We define ordered_square to be OrderedPair unit_interval unit_interval of type set.
Definition. We define ordered_square_topology to be order_topology ordered_square of type set.
Definition. We define ordered_square_open_strip to be ordered_square of type set.
Definition. We define ordered_square_subspace_topology to be subspace_topology (OrderedPair R R) R2_dictionary_order_topology ordered_square of type set.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (ex16_1_subspace_transitive)
∀X Tx Y A : set, topology_on X TxY XA Ysubspace_topology Y (subspace_topology X Tx Y) A = subspace_topology X Tx A
Proof:
The rest of this subproof is missing.
Theorem. (ex16_2_finer_subspaces)
∀X T T' Y : set, topology_on X Ttopology_on X T'T' Tsubspace_topology X T' Y subspace_topology X T Y
Proof:
The rest of this subproof is missing.
Definition. We define interval_A to be open_interval Empty (𝒫 Empty) of type set.
Definition. We define interval_B to be open_interval (𝒫 Empty) (𝒫 (𝒫 Empty)) of type set.
Definition. We define interval_C to be open_interval Empty Empty of type set.
Definition. We define interval_D to be open_interval (𝒫 Empty) (𝒫 Empty) of type set.
Definition. We define interval_E to be open_interval (𝒫 (𝒫 Empty)) (𝒫 (𝒫 Empty)) of type set.
Theorem. (ex16_3_open_sets_subspace)
∀X Tx Y : set, topology_on X TxY X∀U : set, open_in Y (subspace_topology X Tx Y) U∃V : set, open_in X Tx V U = V Y
Proof:
The rest of this subproof is missing.
Definition. We define projection_image1 to be λX Y U ⇒ {xX|∃y : set, OrderedPair x y U} of type setsetsetset.
Definition. We define projection_image2 to be λX Y U ⇒ {yY|∃x : set, OrderedPair x y U} of type setsetsetset.
Theorem. (ex16_4_projections_open)
∀X Tx Y Ty : set, topology_on X Txtopology_on Y Ty∀U : set, U product_topology X Tx Y Tyopen_in X Tx (projection_image1 X Y U) open_in Y Ty (projection_image2 X Y U)
Proof:
The rest of this subproof is missing.
Theorem. (ex16_5a_product_monotone)
∀X T T' Y U U' : set, topology_on X Ttopology_on X T'topology_on Y Utopology_on Y U'T T' U U'product_topology X T Y U product_topology X T' Y U'
Proof:
The rest of this subproof is missing.
Theorem. (ex16_5b_product_converse)
∀X T T' Y U U' : set, topology_on X Ttopology_on X T'topology_on Y Utopology_on Y U'product_topology X T Y U product_topology X T' Y U'T T' U U'
Proof:
The rest of this subproof is missing.
Definition. We define rational_rectangle_basis to be {r𝒫 (OrderedPair R R)|∃a b c d : set, a rational_numbers b rational_numbers c rational_numbers d rational_numbers r = OrderedPair (open_interval a b) (open_interval c d)} of type set.
Proof:
The rest of this subproof is missing.
Definition. We define convex_subset to be λA ⇒ A R ∀x y : set, x Ay Aopen_interval x y A of type setprop.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define interior_of to be λX T A ⇒ {xX|∃U : set, U T x U U A} of type setsetsetset.
Definition. We define closure_of to be λX T A ⇒ {xX|∀U : set, U Tx UU A Empty} of type setsetsetset.
Theorem. (closed_sets_axioms)
∀X T : set, topology_on X Tlet C ≔ {X U|UT} in X C Empty C (∀F : set, F 𝒫 Cintersection_of_family F C) (∀A B : set, A CB CA B C)
Proof:
The rest of this subproof is missing.
Theorem. (closed_in_subspace_iff_intersection)
∀X Tx Y A : set, topology_on X TxY X(closed_in Y (subspace_topology X Tx Y) A ∃C : set, closed_in X Tx C A = C Y)
Proof:
The rest of this subproof is missing.
Theorem. (closed_in_closed_subspace)
∀X Tx Y A : set, topology_on X Txclosed_in X Tx Yclosed_in Y (subspace_topology X Tx Y) Aclosed_in X Tx A
Proof:
The rest of this subproof is missing.
Theorem. (closure_in_subspace)
∀X Tx Y A : set, topology_on X TxY Xclosure_of Y (subspace_topology X Tx Y) A = (closure_of X Tx A) Y
Proof:
The rest of this subproof is missing.
Theorem. (closure_characterization)
∀X Tx A x : set, topology_on X Tx(x closure_of X Tx A (∀UTx, x UU A Empty))
Proof:
The rest of this subproof is missing.
Definition. We define limit_point_of to be λX Tx A x ⇒ topology_on X Tx x X ∀U : set, U Txx U∃y : set, y A y x y U of type setsetsetsetprop.
Definition. We define limit_points_of to be λX Tx A ⇒ {xX|limit_point_of X Tx A x} of type setsetsetset.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Definition. We define Hausdorff_space to be λX Tx ⇒ topology_on X Tx ∀x1 x2 : set, x1 x2∃U V : set, U Tx V Tx x1 U x2 V U V = Empty of type setsetprop.
Definition. We define T1_space to be λX Tx ⇒ topology_on X Tx (∀F : set, finite Fclosed_in X Tx F) of type setsetprop.
Theorem. (finite_sets_closed_in_Hausdorff)
∀X Tx : set, Hausdorff_space X Tx∀F : set, finite Fclosed_in X Tx F
Proof:
The rest of this subproof is missing.
Theorem. (limit_points_infinite_neighborhoods)
∀X Tx A x : set, T1_space X Txlimit_point_of X Tx A x (∀UTx, x Uinfinite (U A))
Proof:
The rest of this subproof is missing.
Theorem. (Hausdorff_unique_limits)
∀X Tx seq x y : set, Hausdorff_space X Txx yfunction_on seq ω X(∀U : set, U Txx U∃N : set, N ω ∀n : set, n ωN napply_fun seq n U)(∀U : set, U Txy U∃N : set, N ω ∀n : set, n ωN napply_fun seq n U)False
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (ex17_1_topology_from_closed_sets)
∀X Tx : set, closed_in X Tx X(∀A : set, closed_in X Tx Aclosed_in X Tx (X A))topology_on X Tx
Proof:
The rest of this subproof is missing.
Theorem. (ex17_2_closed_in_closed_subspace)
∀X Tx Y A : set, closed_in X Tx Yclosed_in Y (subspace_topology X Tx Y) Aclosed_in X Tx A
Proof:
The rest of this subproof is missing.
Theorem. (ex17_3_product_of_closed_sets_closed)
∀X Tx Y Ty A B : set, closed_in X Tx Aclosed_in Y Ty Bclosed_in (OrderedPair X Y) (product_topology X Tx Y Ty) (OrderedPair A B)
Proof:
The rest of this subproof is missing.
Theorem. (ex17_4_open_minus_closed_and_closed_minus_open)
∀X Tx U A : set, topology_on X Txopen_in X Tx Uclosed_in X Tx Aopen_in X Tx (U A) closed_in X Tx (A U)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (ex17_6_closure_properties)
∀X Tx A : set, topology_on X Txclosure_of X Tx (closure_of X Tx A) = closure_of X Tx A closed_in X Tx (closure_of X Tx A)
Proof:
The rest of this subproof is missing.
Theorem. (ex17_7_counterexample_union_closure)
∀X Tx A B : set, topology_on X Txclosed_in X Tx (A B)¬ (closed_in X Tx A closed_in X Tx B)
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Proof:
The rest of this subproof is missing.
Theorem. (ex17_14_sequence_in_finite_complement_topology)
∀X seq : set, function_on seq ω X∀x : set, x X∀U : set, U finite_complement_topology Xx U∃N : set, N ω ∀n : set, n ωN napply_fun seq n U
Proof:
The rest of this subproof is missing.
Theorem. (ex17_15_T1_characterization)
∀X Tx : set, topology_on X Tx(T1_space X Tx (∀x : set, closed_in X Tx {x}))
Proof:
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Proof:
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Proof:
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Proof:
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Definition. We define boundary_of to be λX Tx A ⇒ closure_of X Tx A closure_of X Tx (X A) of type setsetsetset.
Theorem. (ex17_19_boundary_properties)
∀X Tx A : set, topology_on X Txboundary_of X Tx A closure_of X Tx A boundary_of X Tx A closure_of X Tx (X A)
Proof:
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Proof:
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Proof:
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Definition. We define preimage_of to be λX f V ⇒ {xX|apply_fun f x V} of type setsetsetset.
Definition. We define continuous_map to be λX Tx Y Ty f ⇒ topology_on X Tx topology_on Y Ty function_on f X Y ∀V : set, V Typreimage_of X f V Tx of type setsetsetsetsetprop.
Theorem. (continuity_equiv_forms)
∀X Tx Y Ty f : set, topology_on X Txtopology_on Y Ty(continuous_map X Tx Y Ty f (∀V : set, V Typreimage_of X f V Tx) (∀C : set, closed_in Y Ty Cclosed_in X Tx (preimage_of X f C)) (∀x : set, x X∀V : set, V Tyapply_fun f x V∃U : set, U Tx x U ∀u : set, u Uapply_fun f u V))
Proof:
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Theorem. (identity_continuous)
∀X Tx : set, topology_on X Txlet id ≔ {UPair x x|xX} in continuous_map X Tx X Tx id
Proof:
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Theorem. (composition_continuous)
∀X Tx Y Ty Z Tz f g : set, continuous_map X Tx Y Ty fcontinuous_map Y Ty Z Tz gcontinuous_map X Tx Z Tz (Empty)
Proof:
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Theorem. (continuous_construction_rules)
∀X Tx Y Ty Z Tz f g : set, continuous_map X Tx Y Ty fcontinuous_map X Tx Y Ty gcontinuous_map X Tx Y Ty f continuous_map X Tx Y Ty g continuous_map X Tx Y Ty g
Proof:
The rest of this subproof is missing.
Definition. We define homeomorphism to be λX Tx Y Ty f ⇒ continuous_map X Tx Y Ty f ∃g : set, continuous_map Y Ty X Tx g (∀x : set, x Xapply_fun g (apply_fun f x) = x) (∀y : set, y Yapply_fun f (apply_fun g y) = y) of type setsetsetsetsetprop.
Theorem. (continuous_on_subspace)
∀X Tx Y Ty f A : set, topology_on X TxA Xcontinuous_map X Tx Y Ty fcontinuous_map A (subspace_topology X Tx A) Y Ty f
Proof:
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Theorem. (homeomorphism_inverse_continuous)
∀X Tx Y Ty f : set, homeomorphism X Tx Y Ty fcontinuous_map Y Ty X Tx f
Proof:
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Theorem. (pasting_lemma)
∀X A B Y Tx Ty f g : set, topology_on X TxA TxB TxA B = Emptycontinuous_map A (subspace_topology X Tx A) Y Ty fcontinuous_map B (subspace_topology X Tx B) Y Ty gcontinuous_map (A B) (subspace_topology X Tx (A B)) Y Ty (f g)
Proof:
The rest of this subproof is missing.
Theorem. (maps_into_products)
∀A X Tx Y Ty f g : set, continuous_map A Tx X Ty fcontinuous_map A Tx Y Ty gcontinuous_map A Tx (OrderedPair X Y) (product_topology X Ty Y Ty) (f g)
Proof:
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Definition. We define projection_map to be λX Y ⇒ projection1 X Y of type setsetset.
Proof:
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Theorem. (product_topology_universal)
∀X Tx Y Ty : set, topology_on X Txtopology_on Y Ty∃Tprod : set, topology_on (OrderedPair X Y) Tprod continuous_map (OrderedPair X Y) Tprod X Tx (projection_map X Y) continuous_map (OrderedPair X Y) Tprod Y Ty (projection_map Y X)
Proof:
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Definition. We define metric_on to be λX d ⇒ function_on d (OrderedPair X X) R (∀x y : set, x Xy Xapply_fun d (OrderedPair x y) = apply_fun d (OrderedPair y x)) (∀x : set, x Xapply_fun d (OrderedPair x x) = 0) (∀x y : set, x Xy X¬ (Rlt (apply_fun d (OrderedPair x y)) 0) apply_fun d (OrderedPair x y) = 0x = y) (∀x y z : set, x Xy Xz XRlt (apply_fun d (OrderedPair x z)) (apply_fun d (OrderedPair x y) apply_fun d (OrderedPair y z))) of type setsetprop.
Definition. We define open_ball to be λX d x ⇒ {yX|∃rR, Rlt (d x y) r} of type setsetsetset.
Definition. We define metric_topology to be λX d ⇒ generated_topology X {open_ball X d x|xX} of type setsetset.
Proof:
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Proof:
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Proof:
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Theorem. (metric_epsilon_delta_continuity)
∀X dX Y dY f : set, metric_on X dXmetric_on Y dYcontinuous_map X (metric_topology X dX) Y (metric_topology Y dY) f (∀x0 : set, x0 X∀eps : set, eps R Rlt 0 eps∃delta : set, delta R Rlt 0 delta (∀x : set, x XRlt (apply_fun dX (OrderedPair x x0)) deltaRlt (apply_fun dY (OrderedPair (apply_fun f x) (apply_fun f x0))) eps))
Proof:
The rest of this subproof is missing.
Definition. We define sequence_in to be λseq A ⇒ seq A of type setsetprop.
Definition. We define sequence_on to be λseq A ⇒ function_on seq ω A of type setsetprop.
Definition. We define converges_to to be λX Tx seq x ⇒ topology_on X Tx sequence_on seq X x X ∀U : set, U Txx U∃N : set, N ω ∀n : set, n ωN napply_fun seq n U of type setsetsetsetprop.
Definition. We define image_of to be λf seq ⇒ Repl seq (λy ⇒ y) of type setsetset.
Definition. We define function_sequence_value to be λf_seq n x ⇒ apply_fun (apply_fun f_seq n) x of type setsetsetset.
Definition. We define sequence_converges_metric to be λX d seq x ⇒ metric_on X d sequence_on seq X x X ∀eps : set, eps R Rlt 0 eps∃N : set, N ω ∀n : set, n ωN nRlt (apply_fun d (OrderedPair (apply_fun seq n) x)) eps of type setsetsetsetprop.
Theorem. (metric_limits_unique)
∀X d seq x y : set, metric_on X dsequence_on seq Xsequence_converges_metric X d seq xsequence_converges_metric X d seq yx = y
Proof:
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Definition. We define uniform_convergence_functions to be λX dX Y dY f_seq f ⇒ metric_on X dX metric_on Y dY function_on f_seq ω (function_space X Y) function_on f X Y (∀n : set, n ωfunction_on (apply_fun f_seq n) X Y) ∀eps : set, eps R Rlt 0 eps∃N : set, N ω ∀n : set, n ωN n∀x : set, x XRlt (apply_fun dY (OrderedPair (apply_fun (apply_fun f_seq n) x) (apply_fun f x))) eps of type setsetsetsetsetsetprop.
Theorem. (uniform_limit_of_continuous_is_continuous)
∀X dX Y dY f_seq f : set, metric_on X dXmetric_on Y dYfunction_on f_seq ω (function_space X Y)(∀n : set, n ωcontinuous_map X (metric_topology X dX) Y (metric_topology Y dY) (apply_fun f_seq n))uniform_convergence_functions X dX Y dY f_seq fcontinuous_map X (metric_topology X dX) Y (metric_topology Y dY) f
Proof:
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Proof:
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Proof:
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Definition. We define quotient_topology to be λX Tx Y f ⇒ {V𝒫 Y|{xX|apply_fun f x V} Tx} of type setsetsetsetset.
Definition. We define quotient_map to be λX Tx Y f ⇒ topology_on X Tx function_on f X Y (∀y : set, y Y∃x : set, x X apply_fun f x = y) of type setsetsetsetprop.
Proof:
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Theorem. (quotient_universal_property)
∀X Tx Y Ty f : set, quotient_map X Tx Y ftopology_on Y Tycontinuous_map X Tx Y Ty f
Proof:
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Definition. We define separation_of to be λX U V ⇒ U 𝒫 X V 𝒫 X U V = Empty U Empty V Empty of type setsetsetprop.
Definition. We define connected_space to be λX Tx ⇒ topology_on X Tx ¬ (∃U V : set, U Tx V Tx separation_of X U V U V = X) of type setsetprop.
Proof:
The rest of this subproof is missing.
Theorem. (separation_subspace_limit_points)
∀X Tx Y : set, topology_on X Tx∃A B : set, A B = Empty A B = Y open_in X Tx A open_in X Tx B ∃a b : set, limit_point_of X Tx A a limit_point_of X Tx B b a Y b Y
Proof:
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Theorem. (connected_subset_in_separation_side)
∀X Tx C D Y : set, topology_on X Txconnected_space Y TxC D = EmptyC D = Xopen_in X Tx Copen_in X Tx DY C Y D
Proof:
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Theorem. (union_connected_common_point)
∀X Tx F : set, topology_on X Tx(∀C : set, C Fconnected_space C (subspace_topology X Tx C))(∃x : set, ∀C : set, C Fx C)connected_space ( F) (subspace_topology X Tx ( F))
Proof:
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Proof:
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Theorem. (continuous_image_connected)
∀X Tx Y Ty f : set, connected_space X Txcontinuous_map X Tx Y Ty fconnected_space Y Ty
Proof:
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Proof:
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Theorem. (connected_subsets_real_are_intervals)
∀A : set, A Rconnected_space A (subspace_topology R R_standard_topology A)∀x y z : set, x Ay Az R(Rlt x z Rlt z y Rlt y z Rlt z x)z A
Proof:
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Proof:
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Proof:
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Proof:
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Definition. We define path_between to be λX x y p ⇒ function_on p unit_interval X apply_fun p 0 = x apply_fun p 1 = y of type setsetsetsetprop.
Definition. We define path_connected_space to be λX Tx ⇒ topology_on X Tx ∀x y : set, x Xy X∃p : set, path_between X x y p of type setsetprop.
Proof:
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Proof:
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Proof:
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Definition. We define path_component_of to be λX Tx x ⇒ {yX|∃p : set, function_on p unit_interval X continuous_map unit_interval R_standard_topology X Tx p apply_fun p 0 = x apply_fun p 1 = y} of type setsetsetset.
Theorem. (path_components_equivalence_relation)
∀X Tx : set, topology_on X Tx(∀x : set, x Xx path_component_of X Tx x) (∀x y : set, x Xy Xy path_component_of X Tx xx path_component_of X Tx y) (∀x y z : set, x Xy Xz Xy path_component_of X Tx xz path_component_of X Tx yz path_component_of X Tx x)
Proof:
The rest of this subproof is missing.
Definition. We define component_of to be λX Tx x ⇒ {yX|∃C : set, connected_space C (subspace_topology X Tx C) x C y C} of type setsetsetset.
Definition. We define locally_connected to be λX Tx ⇒ topology_on X Tx ∀x : set, x X∀U : set, U Txx U∃V : set, V Tx x V V U connected_space V (subspace_topology X Tx V) of type setsetprop.
Definition. We define locally_path_connected to be λX Tx ⇒ topology_on X Tx ∀x : set, x X∃U : set, U Tx x U path_connected_space U (subspace_topology X Tx U) of type setsetprop.
Definition. We define pairwise_disjoint to be λFam ⇒ ∀U V : set, U FamV FamU VU V = Empty of type setprop.
Definition. We define covers to be λX U ⇒ ∀x : set, x X∃u : set, u U x u of type setsetprop.
Theorem. (path_components_open)
∀X Tx : set, locally_path_connected X Tx∀x : set, x Xopen_in X Tx (path_component_of X Tx x)
Proof:
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Proof:
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Theorem. (components_are_closed)
∀X Tx : set, topology_on X Tx∀x : set, x Xclosed_in X Tx (component_of X Tx x)
Proof:
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Proof:
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Proof:
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Definition. We define quasicomponent_of to be λX Tx x ⇒ {yX|∀U : set, open_in X Tx Uclosed_in X Tx Ux Uy U} of type setsetsetset.
Theorem. (components_vs_quasicomponents)
∀X Tx : set, topology_on X Tx(∀x : set, component_of X Tx x quasicomponent_of X Tx x) (locally_connected X Tx∀x : set, component_of X Tx x = quasicomponent_of X Tx x)
Proof:
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Proof:
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Proof:
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Proof:
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Proof:
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Proof:
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Definition. We define open_cover_of to be λX Tx Fam ⇒ topology_on X Tx Fam 𝒫 X X Fam (∀U : set, U FamU Tx) of type setsetsetprop.
Definition. We define has_finite_subcover to be λX Tx Fam ⇒ ∃G : set, G Fam finite G X G of type setsetsetprop.
Definition. We define compact_space to be λX Tx ⇒ topology_on X Tx ∀Fam : set, open_cover_of X Tx Famhas_finite_subcover X Tx Fam of type setsetprop.
Theorem. (Heine_Borel_subcover)
∀X Tx Fam : set, compact_space X Txopen_cover_of X Tx Famhas_finite_subcover X Tx Fam
Proof:
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Theorem. (compact_subspace_via_ambient_covers)
∀X Tx Y : set, topology_on X Tx(compact_space Y (subspace_topology X Tx Y) ∀Fam : set, open_cover_of Y Tx Famhas_finite_subcover Y Tx Fam)
Proof:
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Proof:
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Proof:
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Theorem. (Hausdorff_separate_point_compact_set)
∀X Tx Y x : set, Hausdorff_space X Txcompact_space Y (subspace_topology X Tx Y)x Y = Empty∃U V : set, U Tx V Tx x U Y V U V = Empty
Proof:
The rest of this subproof is missing.
Theorem. (continuous_image_compact)
∀X Tx Y Ty f : set, compact_space X Txcontinuous_map X Tx Y Ty fcompact_space Y Ty
Proof:
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Theorem. (tube_lemma)
∀X Tx Y Ty : set, topology_on X Txtopology_on Y Tycompact_space X Tx∀x0 : set, x0 X∀N : set, N product_topology X Tx Y Ty x0 N∃U : set, U Tx x0 U (∀y : set, y YOrderedPair U y N)
Proof:
The rest of this subproof is missing.
Definition. We define bijection to be λX Y f ⇒ function_on f X Y (∀y : set, y Y∃x : set, x X apply_fun f x = y (∀x' : set, x' Xapply_fun f x' = yx' = x)) of type setsetsetprop.
Definition. We define Abs to be abs_SNo of type setset.
Theorem. (compact_to_Hausdorff_bijection_homeomorphism)
∀X Tx Y Ty f : set, compact_space X TxHausdorff_space Y Tycontinuous_map X Tx Y Ty fbijection X Y fhomeomorphism X Tx Y Ty f
Proof:
The rest of this subproof is missing.
Definition. We define bounded_subset_of_reals to be λA ⇒ ∃M : set, M R ∀x : set, x A¬ (Rlt M (Abs x)) of type setprop.
Proof:
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Proof:
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Proof:
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Proof:
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Definition. We define limit_point_compact to be λX Tx ⇒ topology_on X Tx ∀A : set, A Xinfinite A∃x : set, limit_point_of X Tx A x of type setsetprop.
Proof:
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Proof:
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Definition. We define locally_compact to be λX Tx ⇒ topology_on X Tx ∀x : set, x X∃U : set, U Tx x U compact_space (closure_of X Tx U) (subspace_topology X Tx (closure_of X Tx U)) of type setsetprop.
Proof:
The rest of this subproof is missing.
Definition. We define one_point_compactification to be λX Tx Y Ty ⇒ compact_space Y Ty Hausdorff_space Y Ty X Y ∃p : set, p Y ¬ p X subspace_topology Y Ty X = Tx (∀y : set, y Yy X y = p) of type setsetsetsetprop.
Proof:
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Proof:
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Definition. We define directed_set to be λJ ⇒ J Empty ∀i j : set, i Jj J∃k : set, k J of type setprop.
Proof:
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Theorem. (cofinal_subset_directed)
∀J K : set, directed_set JK J(∀i : set, i J∃k : set, k K i K i J)directed_set K
Proof:
The rest of this subproof is missing.
Definition. We define net_on to be λnet ⇒ ∃J X : set, directed_set J function_on net J X of type setprop.
Definition. We define subnet_of to be λnet sub ⇒ ∃J X K Y phi : set, directed_set J function_on net J X directed_set K function_on sub K Y function_on phi K J (∀k1 k2 : set, k1 Kk2 K∃k3 : set, k3 K apply_fun phi k3 = apply_fun phi k1 apply_fun phi k3 = apply_fun phi k2) (∀k : set, k K∃j : set, j J apply_fun phi k = j apply_fun sub k = apply_fun net j) of type setsetprop.
Definition. We define accumulation_point_of_net to be λX net x ⇒ ∃J X0 : set, directed_set J function_on net J X0 x X ∀U : set, x U∃i : set, i J apply_fun net i U apply_fun net i x of type setsetsetprop.
Definition. We define net_converges to be λX Tx net x ⇒ ∃J X0 : set, topology_on X Tx directed_set J function_on net J X0 x X ∀U : set, U Txx U∃i : set, i J apply_fun net i U of type setsetsetsetprop.
Theorem. (subnet_preserves_convergence)
∀X Tx net sub x : set, net_converges X Tx net xsubnet_of net subnet_converges X Tx sub x
Proof:
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Theorem. (closure_via_nets)
∀X Tx A x : set, topology_on X Tx(x closure_of X Tx A ∃net : set, net_on net net_converges X Tx net x)
Proof:
The rest of this subproof is missing.
Theorem. (continuity_via_nets)
∀X Tx Y Ty f : set, topology_on X Txtopology_on Y Ty(continuous_map X Tx Y Ty f ∀net : set, net_on net∀x : set, net_converges X Tx net xnet_converges Y Ty net (Empty))
Proof:
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Theorem. (subnet_converges_to_accumulation)
∀X Tx net x : set, accumulation_point_of_net X net x∃sub : set, subnet_of net sub net_converges X Tx sub x
Proof:
The rest of this subproof is missing.
Theorem. (compact_iff_every_net_has_convergent_subnet)
∀X Tx : set, topology_on X Tx(compact_space X Tx ∀net : set, net_on net∃sub x : set, subnet_of net sub net_converges X Tx sub x)
Proof:
The rest of this subproof is missing.
Definition. We define countable_set to be λA ⇒ A ω of type setprop.
Definition. We define countable_subcollection to be λV U ⇒ V U countable_set V of type setsetprop.
Definition. We define countable_index_set to be λI ⇒ I ω of type setprop.
Definition. We define countable_product_component_topology to be λXi i ⇒ apply_fun Xi i of type setsetset.
Definition. We define real_sequences to be 𝒫 R of type set.
Definition. We define uniform_metric_Romega to be Eps_i (λd ⇒ metric_on real_sequences d) of type set.
Definition. We define uniform_topology to be metric_topology real_sequences uniform_metric_Romega of type set.
Definition. We define open_cover to be λX Tx U ⇒ (∀u : set, u Uu Tx) covers X U of type setsetsetprop.
Definition. We define Lindelof_space to be λX Tx ⇒ topology_on X Tx ∀U : set, open_cover X Tx U∃V : set, countable_subcollection V U covers X V of type setsetprop.
Definition. We define Sorgenfrey_line to be R of type set.
Definition. We define Sorgenfrey_topology to be R_lower_limit_topology of type set.
Definition. We define countable_basis_at to be λX Tx x ⇒ topology_on X Tx ∃B : set, basis_on X B countable_set B (∀U : set, U Txx U∃b : set, b B x b b U) of type setsetsetprop.
Definition. We define first_countable_space to be λX Tx ⇒ topology_on X Tx ∀x : set, x Xcountable_basis_at X Tx x of type setsetprop.
Theorem. (first_countable_sequences_detect_closure)
∀X Tx A x : set, topology_on X Tx(∃seq : set, sequence_in seq A converges_to X Tx seq x)x closure_of X Tx A
Proof:
The rest of this subproof is missing.
Theorem. (first_countable_sequences_detect_continuity)
∀X Tx Y Ty f : set, topology_on X Txtopology_on Y Ty(continuous_map X Tx Y Ty f∀seq : set, sequence_in seq Xconverges_to X Tx seq (Empty)converges_to Y Ty (image_of f seq) f)
Proof:
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Definition. We define second_countable_space to be λX Tx ⇒ topology_on X Tx ∃B : set, basis_on X B countable_set B basis_generates X B Tx of type setsetprop.
Proof:
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Proof:
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Proof:
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Definition. We define dense_in to be λA X Tx ⇒ closure_of X Tx A = X of type setsetsetprop.
Theorem. (countable_basis_implies_Lindelof)
∀X Tx : set, topology_on X Txsecond_countable_space X Tx∀U : set, open_cover X Tx U∃V : set, countable_subcollection V U covers X V
Proof:
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Proof:
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Proof:
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Definition. We define Sorgenfrey_plane_topology to be product_topology Sorgenfrey_line Sorgenfrey_topology Sorgenfrey_line Sorgenfrey_topology of type set.
Definition. We define one_point_sets_closed to be λX Tx ⇒ topology_on X Tx ∀x : set, x Xclosed_in X Tx {x} of type setsetprop.
Definition. We define Hausdorff_spaces_family to be λI Xi ⇒ ∀i : set, i IHausdorff_space (product_component Xi i) (product_component_topology Xi i) of type setsetprop.
Definition. We define regular_spaces_family to be λI Xi ⇒ ∀i : set, i Itopology_on (product_component Xi i) (product_component_topology Xi i) of type setsetprop.
Definition. We define uncountable_set to be λX ⇒ ¬ countable_set X of type setprop.
Definition. We define well_ordered_set to be λX ⇒ ∃alpha : set, ordinal alpha equip X alpha of type setprop.
Definition. We define completely_regular_spaces_family to be λI Xi ⇒ ∀i : set, i Itopology_on (product_component Xi i) (product_component_topology Xi i) of type setsetprop.
Definition. We define separating_family_of_functions to be λX Tx F J ⇒ topology_on X Tx F function_space X J (∀x1 x2 : set, x1 Xx2 Xx1 x2∃f : set, f F apply_fun f x1 apply_fun f x2) of type setsetsetsetprop.
Definition. We define embedding_of to be λX Tx Y Ty f ⇒ function_on f X Y continuous_map X Tx Y Ty f (∀x1 x2 : set, x1 Xx2 Xapply_fun f x1 = apply_fun f x2x1 = x2) of type setsetsetsetsetprop.
Definition. We define power_real to be λJ ⇒ function_space J R of type setset.
Definition. We define unit_interval_power to be λJ ⇒ function_space J unit_interval of type setset.
Definition. We define metrizable to be λX Tx ⇒ ∃d : set, metric_on X d metric_topology X d = Tx of type setsetprop.
Proof:
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Proof:
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Definition. We define regular_space to be λX Tx ⇒ topology_on X Tx ∀x : set, x X∀F : set, closed_in X Tx Fx F∃U V : set, U Tx V Tx x U F V U V = Empty of type setsetprop.
Definition. We define normal_space to be λX Tx ⇒ topology_on X Tx ∀A B : set, closed_in X Tx Aclosed_in X Tx BA B = Empty∃U V : set, U Tx V Tx A U B V U V = Empty of type setsetprop.
Theorem. (regular_normal_via_closure)
∀X Tx : set, topology_on X Tx(one_point_sets_closed X Tx(regular_space X Tx ∀x U : set, x XU Txx U∃V : set, V Tx x V closure_of X Tx V U)) (one_point_sets_closed X Tx(normal_space X Tx ∀A U : set, closed_in X Tx AU TxA U∃V : set, V Tx A V closure_of X Tx V U))
Proof:
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Theorem. (separation_axioms_subspace_product)
∀X Tx : set, topology_on X Tx(∀Y : set, Y XHausdorff_space X TxHausdorff_space Y (subspace_topology X Tx Y)) (∀I Xi : set, Hausdorff_spaces_family I XiHausdorff_space (product_space I Xi) (product_topology_full I Xi)) (∀Y : set, Y Xregular_space X Txregular_space Y (subspace_topology X Tx Y)) (∀I Xi : set, regular_spaces_family I Xiregular_space (product_space I Xi) (product_topology_full I Xi))
Proof:
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Definition. We define R_K to be R of type set.
Proof:
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Proof:
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Proof:
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Definition. We define S_Omega to be ω of type set.
Definition. We define Sbar_Omega to be 𝒫 ω of type set.
Definition. We define SOmega_topology to be discrete_topology S_Omega of type set.
Definition. We define SbarOmega_topology to be discrete_topology Sbar_Omega of type set.
Proof:
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Definition. We define closed_interval to be λa b ⇒ {xR|¬ (Rlt x a) ¬ (Rlt b x)} of type setsetset.
Theorem. (Urysohn_lemma)
∀X Tx A B a b : set, normal_space X Txclosed_in X Tx Aclosed_in X Tx BA B = Empty∃f : set, continuous_map X Tx (closed_interval a b) (order_topology (closed_interval a b)) f
Proof:
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Definition. We define completely_regular_space to be λX Tx ⇒ topology_on X Tx ∀x : set, x X∀F : set, closed_in X Tx Fx F∃f : set, continuous_map X Tx R R_standard_topology f apply_fun f x = 0 ∀y : set, y Fapply_fun f y = 1 of type setsetprop.
Definition. We define Tychonoff_space to be λX Tx ⇒ completely_regular_space X Tx Hausdorff_space X Tx of type setsetprop.
Proof:
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Proof:
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Proof:
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Theorem. (embedding_via_functions)
∀X Tx : set, topology_on X Txone_point_sets_closed X Tx∀F J : set, separating_family_of_functions X Tx F J∃Fmap : set, embedding_of X Tx (power_real J) (product_topology_full J (const_family J R)) Fmap
Proof:
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Proof:
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Theorem. (Tietze_extension_interval)
∀X Tx A a b f : set, normal_space X Txclosed_in X Tx Acontinuous_map A (subspace_topology X Tx A) (closed_interval a b) (order_topology (closed_interval a b)) f∃g : set, continuous_map X Tx (closed_interval a b) (order_topology (closed_interval a b)) g (∀x : set, x Aapply_fun g x = apply_fun f x)
Proof:
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Theorem. (Tietze_extension_real)
∀X Tx A f : set, normal_space X Txclosed_in X Tx Acontinuous_map A (subspace_topology X Tx A) R R_standard_topology f∃g : set, continuous_map X Tx R R_standard_topology g (∀x : set, x Aapply_fun g x = apply_fun f x)
Proof:
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Definition. We define m_manifold to be λX Tx ⇒ Hausdorff_space X Tx second_countable_space X Tx of type setsetprop.
Definition. We define partition_of_unity_dominated to be λX Tx U ⇒ topology_on X Tx open_cover X Tx U ∃P : set, P function_space X R (∀f : set, f Pcontinuous_map X Tx R R_standard_topology f) (∀x : set, x X∃F : set, finite F F P (∀f : set, f Papply_fun f x 0f F) (∀f : set, f F∃u : set, u U {yX|apply_fun f y 0} u)) of type setsetsetprop.
Proof:
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Proof:
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Proof:
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Definition. We define Stone_Cech_compactification to be λX Tx ⇒ {p𝒫 (𝒫 (𝒫 X))|∃Y Ty e : set, p = OrderedPair (OrderedPair Y Ty) e compact_space Y Ty Hausdorff_space Y Ty embedding_of X Tx Y Ty e} of type setsetset.
Proof:
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Definition. We define refine_of to be λV U ⇒ ∀v : set, v V∃u : set, u U v u of type setsetprop.
Definition. We define locally_finite_family to be λX Tx F ⇒ topology_on X Tx ∀x : set, x X∃N : set, N Tx x N ∃S : set, finite S S F ∀A : set, A FA N EmptyA S of type setsetsetprop.
Definition. We define locally_finite_basis to be λX Tx ⇒ topology_on X Tx ∃B : set, basis_on X B locally_finite_family X Tx B of type setsetprop.
Definition. We define sigma_locally_finite_basis to be λX Tx ⇒ topology_on X Tx ∃Fams : set, countable_set Fams Fams 𝒫 (𝒫 X) (∀F : set, F Famslocally_finite_family X Tx F) basis_on X ( Fams) ∀b : set, b Famsb Tx of type setsetprop.
Proof:
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Definition. We define paracompact_space to be λX Tx ⇒ topology_on X Tx ∀U : set, open_cover X Tx U∃V : set, open_cover X Tx V locally_finite_family X Tx V refine_of V U of type setsetprop.
Theorem. (locally_finite_refinement)
∀X Tx U : set, paracompact_space X Txopen_cover X Tx U∃V : set, open_cover X Tx V locally_finite_family X Tx V
Proof:
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Proof:
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Proof:
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Definition. We define cauchy_sequence to be λX d seq ⇒ metric_on X d seq X ∀eps : set, eps R∃N : set, N ω ∀m n : set, m ωn ωN ωRlt (d (apply_fun seq m) (apply_fun seq n)) eps of type setsetsetprop.
Definition. We define complete_metric_space to be λX d ⇒ metric_on X d ∀seq : set, seq Xcauchy_sequence X d seq∃x : set, converges_to X (metric_topology X d) seq x of type setsetprop.
Definition. We define discrete_metric to be λX ⇒ {pOrderedPair X X|∃x : set, ∃y : set, x X y X ((x = y p = OrderedPair (OrderedPair x y) 0) (x y p = OrderedPair (OrderedPair x y) 1))} of type setset.
Definition. We define euclidean_metric to be λn ⇒ discrete_metric (euclidean_space n) of type setset.
Definition. We define bounded_product_metric to be λJ ⇒ discrete_metric (power_real J) of type setset.
Theorem. (Cauchy_with_convergent_subsequence_converges)
∀X d seq x : set, metric_on X dcauchy_sequence X d seq(∃subseq : set, subseq seq converges_to X (metric_topology X d) subseq x)converges_to X (metric_topology X d) seq x
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Proof:
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Definition. We define unit_square to be OrderedPair unit_interval unit_interval of type set.
Definition. We define unit_square_topology to be product_topology unit_interval R_standard_topology unit_interval R_standard_topology of type set.
Proof:
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Definition. We define sequentially_compact to be λX Tx ⇒ topology_on X Tx ∀seq : set, seq X∃x : set, converges_to X Tx seq x of type setsetprop.
Proof:
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Definition. We define pointwise_convergence_topology to be λX Tx Y Ty ⇒ generated_topology (function_space X Y) Empty of type setsetsetsetset.
Definition. We define compact_convergence_topology to be λX Tx Y Ty ⇒ generated_topology (function_space X Y) Empty of type setsetsetsetset.
Definition. We define equicontinuous_family to be λX Tx Y Ty F ⇒ topology_on X Tx topology_on Y Ty F function_space X Y ∀x : set, x X∀V : set, V Ty(∃f0 : set, f0 F apply_fun f0 x V)∃U : set, U Tx x U ∀f : set, f F∀y : set, y Uapply_fun f y V of type setsetsetsetsetprop.
Definition. We define relatively_compact_in_compact_convergence to be λX Tx Y Ty F ⇒ topology_on X Tx topology_on Y Ty F function_space X Y compact_space F (compact_convergence_topology X Tx Y Ty) of type setsetsetsetsetprop.
Proof:
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Definition. We define intersection_over_family to be λX Fam ⇒ {xX|∀U : set, U Famx U} of type setsetset.
Definition. We define Baire_space to be λTx ⇒ ∃X : set, topology_on X Tx ∀U : set, U Txcountable_set U(∀u : set, u Uu Tx dense_in u X Tx)dense_in (intersection_over_family X U) X Tx of type setprop.
Theorem. (Baire_space_dense_Gdelta)
∀Tx : set, (Baire_space Tx ∃X : set, topology_on X Tx ∀U : set, U Txcountable_set U(∀u : set, u Uu Tx dense_in u X Tx)dense_in (intersection_over_family X U) X Tx)
Proof:
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Proof:
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Proof:
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Theorem. (Baire_category_theorem)
∀X : set, Baire_space X∀U : set, open_in X X UU Empty
Proof:
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Definition. We define differentiable_at to be λf x ⇒ False of type setsetprop.
Definition. We define nowhere_differentiable to be λf ⇒ function_on f R R ∀x : set, x R¬ differentiable_at f x of type setprop.
Proof:
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Definition. We define cardinality_exact to be λS n ⇒ ordinal n equip S n of type setsetprop.
Definition. We define cardinality_at_most to be λS n ⇒ ordinal n ∃k : set, ordinal k k n equip S k of type setsetprop.
Definition. We define collection_has_order_at_m_plus_one to be λX A m ⇒ ordinal m (∃x : set, x X ∃Fam : set, Fam A finite Fam cardinality_exact Fam m ∀U : set, U Famx U) ∀x : set, x Xcardinality_at_most {UA|x U} m of type setsetsetprop.
Definition. We define covering_dimension to be λX n ⇒ n ω ∃Tx : set, topology_on X Tx of type setsetprop.
Definition. We define finite_dimensional_space to be λX Tx ⇒ topology_on X Tx ∃m : set, covering_dimension X m of type setsetprop.
Proof:
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Proof:
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Proof:
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Proof:
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Theorem. (dimension_finite_union_closed_max)
∀X Fam n : set, finite Fam(∀Y : set, Y Famcovering_dimension Y n)covering_dimension ( Fam) n
Proof:
The rest of this subproof is missing.
Definition. We define ex30_1_Gdelta_points to be ω of type set.
Definition. We define ex30_2_basis_contains_countable to be ω of type set.
Definition. We define ex30_3_uncountably_many_limit_points to be ω of type set.
Definition. We define ex30_4_compact_metrizable_second_countable to be ω of type set.
Definition. We define ex30_5_metrizable_density_Lindelof_imply_second_countable to be ω of type set.
Definition. We define ex30_6_Sorgenfrey_and_ordered_square_not_metrizable to be ω of type set.
Definition. We define ex30_7_SOmega_countability_axioms to be ω of type set.
Definition. We define ex30_8_Romega_uniform_countability to be ω of type set.
Definition. We define ex30_9_closed_Lindelof_and_dense_subsets to be ω of type set.
Definition. We define ex30_10_product_countable_dense to be ω of type set.
Definition. We define ex30_11_image_preserves_Lindelof_or_dense to be ω of type set.
Definition. We define ex30_12_open_map_preserves_countability_axioms to be ω of type set.
Definition. We define ex30_13_disjoint_open_sets_countable to be ω of type set.
Definition. We define ex30_14_product_Lindelof_compact to be ω of type set.
Definition. We define ex30_15_CI_has_countable_dense_uniform to be ω of type set.
Definition. We define ex30_16_product_RI_dense_subset_cardinality to be ω of type set.
Definition. We define ex30_17_Romega_box_countability to be ω of type set.
Definition. We define ex30_18_first_countable_group_countable_basis to be ω of type set.
Definition. We define ex31_1_regular_disjoint_closure_neighborhoods to be ω of type set.
Definition. We define ex31_2_normal_disjoint_closure_neighborhoods to be ω of type set.
Definition. We define ex31_3_order_topology_regular to be ω of type set.
Definition. We define ex31_4_comparison_topologies_separation to be ω of type set.
Definition. We define ex31_5_equalizer_closed_in_Hausdorff to be ω of type set.
Definition. We define ex31_6_closed_map_preserves_normal to be ω of type set.
Definition. We define ex31_7_perfect_map_properties to be ω of type set.
Definition. We define ex31_8_orbit_space_properties to be ω of type set.
Definition. We define ex31_9_Sorgenfrey_plane_no_separation to be ω of type set.
Definition. We define ex32_1_closed_subspace_normal to be ω of type set.
Definition. We define ex32_2_factors_inherit_separation to be ω of type set.
Definition. We define ex32_3_locally_compact_Hausdorff_regular to be ω of type set.
Definition. We define ex32_4_regular_Lindelof_normal to be ω of type set.
Definition. We define ex32_5_Romega_normality_questions to be ω of type set.
Definition. We define ex32_6_completely_normal_characterization to be ω of type set.
Definition. We define ex32_7_completely_normal_examples to be ω of type set.
Definition. We define ex32_8_linear_continuum_normal to be ω of type set.
Definition. We define ex32_9_uncountable_product_not_normal to be ω of type set.
Definition. We define Gdelta_in to be λX Tx A ⇒ ∃Fam : set, countable Fam (∀UFam, open_in X Tx U) Intersection_Fam Fam = A of type setsetsetprop.
Definition. We define perfectly_normal_space to be λX Tx ⇒ normal_space X Tx (∀A : set, closed_in X Tx AGdelta_in X Tx A) of type setsetprop.
Definition. We define separated_subsets to be λX Tx A B ⇒ closure_of X Tx A B = Empty A closure_of X Tx B = Empty of type setsetsetsetprop.
Definition. We define completely_normal_space to be λX Tx ⇒ normal_space X Tx (∀A B : set, separated_subsets X Tx A B∃U V : set, open_in X Tx U open_in X Tx V A U B V U V = Empty) of type setsetprop.
Definition. We define topological_group to be λG Tg ⇒ topology_on G Tg ∃mult inv e : set, function_on mult (OrderedPair G G) G function_on inv G G e G continuous_map (OrderedPair G G) (product_topology G Tg G Tg) G Tg mult continuous_map G Tg G Tg inv of type setsetprop.
Definition. We define ex33_1_level_sets_urysohn to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A B f : set, p = OrderedPair (OrderedPair (OrderedPair X Tx) (OrderedPair A B)) f normal_space X Tx closed_in X Tx A closed_in X Tx B A B = Empty function_on f X R continuous_map X Tx R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x Bapply_fun f x = 1)} of type set.
Definition. We define ex33_2_connected_normal_regular_uncountable to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx connected_space X Tx normal_space X Tx regular_space X Tx uncountable_set X} of type set.
Definition. We define ex33_3_urysohn_metric_direct to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X d A B f : set, p = OrderedPair (OrderedPair (OrderedPair X d) (OrderedPair A B)) f metric_on X d closed_in X (metric_topology X d) A closed_in X (metric_topology X d) B A B = Empty function_on f X R continuous_map X (metric_topology X d) R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x Bapply_fun f x = 1)} of type set.
Definition. We define ex33_4_closed_Gdelta_vanishing_function to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx F f : set, p = OrderedPair (OrderedPair (OrderedPair X Tx) F) f normal_space X Tx closed_in X Tx F Gdelta_in X Tx F function_on f X R continuous_map X Tx R R_standard_topology f (∀x : set, x Fapply_fun f x = 0)} of type set.
Definition. We define ex33_5_strong_urysohn to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A B f : set, p = OrderedPair (OrderedPair (OrderedPair X Tx) (OrderedPair A B)) f normal_space X Tx closed_in X Tx A closed_in X Tx B A B = Empty function_on f X (closed_interval 0 1) continuous_map X Tx R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x Bapply_fun f x = 1)} of type set.
Definition. We define ex33_6_perfect_normality to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx perfectly_normal_space X Tx} of type set.
Definition. We define ex33_7_locally_compact_Hausdorff_completely_regular to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx locally_compact X Tx Hausdorff_space X Tx completely_regular_space X Tx} of type set.
Definition. We define ex33_8_compact_subset_continuous_separation to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A B f : set, p = OrderedPair (OrderedPair (OrderedPair X Tx) (OrderedPair A B)) f normal_space X Tx compact_space A (subspace_topology X Tx A) closed_in X Tx B A B = Empty function_on f X R continuous_map X Tx R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x Bapply_fun f x = 1)} of type set.
Definition. We define ex33_9_Romega_box_completely_regular to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx X = product_space ω (const_family ω R) Tx = box_topology ω (const_family ω R) completely_regular_space X Tx} of type set.
Definition. We define ex33_10_topological_group_completely_regular to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃G Tg : set, p = OrderedPair G Tg topological_group G Tg completely_regular_space G Tg} of type set.
Definition. We define ex33_11_regular_not_completely_regular to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx regular_space X Tx ¬ completely_regular_space X Tx} of type set.
Definition. We define locally_metrizable_space to be λX Tx ⇒ topology_on X Tx ∀x : set, x X∃N : set, N Tx x N ∃d : set, metric_on N d subspace_topology X Tx N = metric_topology N d of type setsetprop.
Definition. We define retraction_of to be λX Tx A ⇒ A X ∃r : set, function_on r X X continuous_map X Tx X Tx r (∀x : set, x Xapply_fun r x A) (∀x : set, x Aapply_fun r x = x) of type setsetsetprop.
Definition. We define image_of_map to be λX Tx Y Ty f ⇒ {apply_fun f x|xX} of type setsetsetsetsetset.
Definition. We define absolute_retract to be λX Tx ⇒ Hausdorff_space X Tx ∀Y Ty, normal_space Y Ty∃e : set, embedding_of X Tx Y Ty e ∃r : set, retraction_of Y Ty (image_of_map X Tx Y Ty e) of type setsetprop.
Definition. We define coherent_topology to be λX Tx Y Ty ⇒ topology_on X Tx topology_on Y Ty X Y subspace_topology Y Ty X = Tx of type setsetsetsetprop.
Definition. We define compact_spaces_family to be λI Xi ⇒ ∀i : set, i Icompact_space (product_component Xi i) (product_component_topology Xi i) of type setsetprop.
Definition. We define surjective_map to be λX Y f ⇒ function_on f X Y ∀y : set, y Y∃x : set, x X apply_fun f x = y of type setsetsetprop.
Definition. We define ex34_1_Hausdorff_countable_basis_not_metrizable_example to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx Hausdorff_space X Tx second_countable_space X Tx ¬ metrizable X Tx} of type set.
Definition. We define ex34_2_completely_normal_not_metrizable_example to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx completely_normal_space X Tx ¬ metrizable X Tx} of type set.
Definition. We define ex34_3_compact_Hausdorff_metrizable_iff_second_countable to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx compact_space X Tx Hausdorff_space X Tx (metrizable X Tx second_countable_space X Tx)} of type set.
Definition. We define ex34_4_locally_compact_Hausdorff_metrizable_questions to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx locally_compact X Tx Hausdorff_space X Tx (second_countable_space X Txmetrizable X Tx)} of type set.
Definition. We define ex34_5_one_point_compactification_metrizable_questions to be {q𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty p : set, q = OrderedPair (OrderedPair (OrderedPair X Tx) (OrderedPair Y Ty)) p one_point_compactification X Tx Y Ty p Y ¬ p X (metrizable X Tx metrizable Y Ty)} of type set.
Definition. We define ex34_6_check_imbedding_proof to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx f : set, p = OrderedPair (OrderedPair X Tx) f completely_regular_space X Tx Hausdorff_space X Tx embedding_of X Tx (power_real ω) (product_topology_full ω (const_family ω R)) f} of type set.
Definition. We define ex34_7_locally_metrizable_compact_Hausdorff_metrizable to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx locally_metrizable_space X Tx compact_space X Tx Hausdorff_space X Tx metrizable X Tx} of type set.
Definition. We define ex34_8_regular_Lindelof_locally_metrizable_metrizable to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx (regular_space X Tx Lindelof_space X Tx locally_metrizable_space X Txmetrizable X Tx)} of type set.
Definition. We define ex34_9_compact_union_two_metrizable_closed_metrizable to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A B : set, p = OrderedPair (OrderedPair X Tx) (OrderedPair A B) compact_space X Tx Hausdorff_space X Tx closed_in X Tx A closed_in X Tx B (UPair A B) = X metrizable A (subspace_topology X Tx A) metrizable B (subspace_topology X Tx B) metrizable X Tx} of type set.
Definition. We define ex35_1_Tietze_implies_Urysohn to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx normal_space X Tx (∀A B : set, closed_in X Tx A closed_in X Tx B A B = Empty∃f : set, continuous_map X Tx R R_standard_topology f (∀x : set, x Aapply_fun f x = 0) (∀x : set, x Bapply_fun f x = 1))} of type set.
Definition. We define ex35_2_interval_partition_parameter to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx normal_space X Tx} of type set.
Definition. We define ex35_3_boundedness_equivalences_metrizable to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx d : set, p = OrderedPair (OrderedPair X Tx) d metric_on X d metric_topology X d = Tx} of type set.
Definition. We define ex35_4_retract_properties to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A : set, p = OrderedPair (OrderedPair X Tx) A retraction_of X Tx A} of type set.
Definition. We define ex35_5_universal_extension_retracts to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A : set, p = OrderedPair (OrderedPair X Tx) A normal_space X Tx retraction_of X Tx A ∀Y Ty f : set, continuous_map A (subspace_topology X Tx A) Y Ty f∃g : set, continuous_map X Tx Y Ty g ∀x : set, x Aapply_fun g x = apply_fun f x} of type set.
Definition. We define ex35_6_absolute_retract_universal_extension to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx absolute_retract X Tx} of type set.
Definition. We define ex35_7_retract_examples to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A : set, p = OrderedPair (OrderedPair X Tx) A retraction_of X Tx A} of type set.
Definition. We define ex35_8_absolute_retract_equivalence to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx absolute_retract X Tx} of type set.
Definition. We define ex35_9_coherent_topology_normal to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty : set, p = OrderedPair (OrderedPair X Tx) (OrderedPair Y Ty) (topology_on X Tx topology_on Y Ty coherent_topology X Tx Y Tynormal_space Y Ty)} of type set.
Definition. We define ex36_manifold_embedding_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃M TM f : set, p = OrderedPair (OrderedPair M TM) f m_manifold M TM∃n : set, embedding_of M TM (euclidean_space n) (euclidean_topology n) f} of type set.
Definition. We define ex37_tychonoff_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃I Xi : set, p = OrderedPair I Xi compact_spaces_family I Xi compact_space (product_space I Xi) (product_topology_full I Xi)} of type set.
Definition. We define ex38_stone_cech_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty : set, p = OrderedPair (OrderedPair X Tx) (OrderedPair Y Ty) completely_regular_space X Tx compact_space Y Ty Hausdorff_space Y Ty ∃e : set, embedding_of X Tx Y Ty e} of type set.
Definition. We define ex39_local_finiteness_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx U : set, p = OrderedPair (OrderedPair X Tx) U locally_finite_family X Tx U} of type set.
Definition. We define ex40_nagata_smirnov_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx B : set, p = OrderedPair (OrderedPair X Tx) B (regular_space X Tx basis_on X B locally_finite_family X Tx Bmetrizable X Tx)} of type set.
Definition. We define ex41_paracompactness_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx U : set, p = OrderedPair (OrderedPair X Tx) U paracompact_space X Tx open_cover X Tx U} of type set.
Definition. We define ex42_smirnov_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx B : set, p = OrderedPair (OrderedPair X Tx) B (regular_space X Tx basis_on X B locally_finite_family X Tx Bmetrizable X Tx)} of type set.
Definition. We define ex43_complete_metric_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X d Tx : set, p = OrderedPair (OrderedPair X d) Tx metric_on X d Tx = metric_topology X d complete_metric_space X d} of type set.
Definition. We define ex44_space_filling_exercises to be {f𝒫 (𝒫 (𝒫 R))|continuous_map unit_interval R2_standard_topology unit_square unit_square_topology f surjective_map unit_interval unit_square f} of type set.
Definition. We define ex45_compact_metric_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X d Tx : set, p = OrderedPair (OrderedPair X d) Tx metric_on X d Tx = metric_topology X d compact_space X Tx} of type set.
Definition. We define ex46_convergence_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty : set, p = OrderedPair (OrderedPair X Tx) (OrderedPair Y Ty) topology_on X Tx topology_on Y Ty True} of type set.
Definition. We define ex47_ascoli_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty : set, p = OrderedPair (OrderedPair X Tx) (OrderedPair Y Ty) compact_space X Tx Hausdorff_space Y Ty} of type set.
Definition. We define ex48_baire_exercises to be {Tx𝒫 (𝒫 R)|Baire_space Tx} of type set.
Definition. We define ex49_nowhere_differentiable_exercises to be {f𝒫 (𝒫 R)|continuous_map R R_standard_topology R R_standard_topology f nowhere_differentiable f} of type set.
Definition. We define ex50_dimension_exercises to be {p𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx n : set, p = OrderedPair (OrderedPair X Tx) n topology_on X Tx ordinal n} of type set.